2021, Md Masud Rana

New Models and Numerical Methods for Partial Differential Equations Applied toSpatial Stoichiometric Aquatic Ecosystems

by

Md Masud Rana

A Dissertation

In

Mathematics and Statistics

Submitted to the Graduate Facultyof Texas Tech University in

Partial Fulfillment ofthe Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Katharine R. LongChair of Committee

Angela PeaceCo-chair of Committee

Victoria E. HowleCo-chair of Committee

Linda J. S. Allen

Chris J. Monico

Mark A. SheridanDean of the Graduate School

August, 2021

Texas Tech University, Md Masud Rana, August 2021

ACKNOWLEDGEMENTS

I wish to express my deepest appreciation to my advisors, Dr. Katharine Long,

Dr. Victoria Howle, and Dr. Angela Peace, for their invaluable advice, guidance,

encouragement, and continuous support throughout my PhD study. Without their

immense help, this dissertation would not have been possible. Their extensive knowl-

edge, profound belief in my abilities, and unparalleled support have taught me a great

deal about scientific research and life in general.

I would like to extend my sincere gratitude to the committee members, Dr. Linda

Allen and Dr. Chris Monico, for their insightful comments and suggestions. I would

also like to acknowledge Dr. Lourdes Juan with whom I have had the pleasure to

work in several projects.

I am grateful to all my friends who have helped me and supported me through-

out this journey. Many thanks to all my classmates at Texas Tech University and

Bangladeshi friends and families in Lubbock for making my graduate life enjoyable.

Finally, I would like to say a heartfelt thank you to my parents and my brothers

and sisters for always believing in me and encouraging me to follow my dream. Most

importantly, I wish to thank my loving and supportive wife, Mukta, and our son,

Aryaav, who provide unending inspiration.

ii


DEDICATION

To my mother who could not see this dissertation completed.

iii


TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Space-Time Population Model . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Genotypic Selection Model . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Model development . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2.1 Spatially heterogeneous model . . . . . . . . . . . . . 11

2.1.2.2 Light absorption . . . . . . . . . . . . . . . . . . . . 12

2.1.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . 15

2.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.4.1 Spatially homogeneous results . . . . . . . . . . . . . 17

2.1.4.2 Spatially heterogeneous results . . . . . . . . . . . . 21

2.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Space-Time Mechanistic Model . . . . . . . . . . . . . . . . . . 27

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2 Model development . . . . . . . . . . . . . . . . . . . . . . 29

2.2.2.1 Light absorption . . . . . . . . . . . . . . . . . . . . 33

2.2.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . 35

iv


2.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3. Reduced-Order Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Description of ROM . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1 The ROM basis and reconstruction of results . . . . . . . . 44

3.2.1.1 An inner-product and norm appropriate to FEM . . 45

3.2.1.2 Finding a reduced-order basis . . . . . . . . . . . . . 46

3.2.1.3 Reduced-order reconstruction of existing results . . . 47

3.2.1.4 Error estimates . . . . . . . . . . . . . . . . . . . . . 48

3.2.2 From a PDE to a low-dimensional ODE system . . . . . . 49

3.2.3 An optimal basis for least squares approximation . . . . . 51

3.2.3.1 Which basis works best for all snapshots? . . . . . . 51

3.2.4 Beyond the simplest cases . . . . . . . . . . . . . . . . . . 54

3.2.4.1 Complex geometry in multiple spatial dimensions . . 54

3.2.4.2 Snapshots from runs with multiple parameters . . . . 54

3.2.5 Non-homogeneous boundary conditions . . . . . . . . . . . 55

3.2.5.1 BCs independent of time and parameters . . . . . . . 55

3.2.5.2 BCs dependent on parameters, independent of time . 55

3.2.5.3 BCs dependent on parameters and time . . . . . . . 56

3.2.6 Multiple unknowns . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Mechanistic Model: ROM Approach . . . . . . . . . . . . . . . 57

3.3.1 ROM basis for mechanistic model . . . . . . . . . . . . . . 57

3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

v


3.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4. Preconditioning Implicit Runge–Kutta Methods for Parabolic PDEs . . 63

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Appropriate Time-Integrator for Parabolic PDEs . . . . . . . . 64

4.2.1 Stability of time-integrator . . . . . . . . . . . . . . . . . . 65

4.2.2 CFL condition . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.3 Second Dahlquist barrier . . . . . . . . . . . . . . . . . . . 67

4.3 Implicit Runge–Kutta Methods . . . . . . . . . . . . . . . . . 68

4.4 Choice of Solvers . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 IRK Methods Applied to Test PDEs . . . . . . . . . . . . . . . 72

4.6.1 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.6.2 Double-Glazing advection-diffusion problem . . . . . . . . 74

4.7 Block Preconditioning IRK Methods for Parabolic PDEs . . . 75

4.7.1 LDU-based block triangular preconditioners . . . . . . . . 77

4.7.2 Application of the preconditioners . . . . . . . . . . . . . . 79

4.7.3 Analysis of the preconditioners . . . . . . . . . . . . . . . . 80

4.7.4 Comparison between PGSL with optimal coefficients and

PLD preconditioners . . . . . . . . . . . . . . . . . . . . . 82

4.8 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . 86

4.8.1 Heat Equation Results . . . . . . . . . . . . . . . . . . . . 89

4.8.1.1 Comparison results between PJ , PGSL, PDU , and PLD

preconditioners . . . . . . . . . . . . . . . . . . . . . 89

vi


4.8.1.2 Comparison results between PGSL with optimal coef-

ficients and PLD preconditioners . . . . . . . . . . . 92

4.8.2 Advection-Diffusion Equation Results . . . . . . . . . . . . 97

4.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

vii


ABSTRACT

Ecological stoichiometry is a framework to study population dynamics that incor-

porates energy flow in trophic levels as well as chemical imbalances. Spatial variation

in an ecological interaction also has effects on the population dynamics. We develop

and analyze numerically two stoichiometric producer-grazer models in a spatially het-

erogeneous environment: a model with two grazing genotypes to understand selection

processes and a mechanistic model that tracks explicitly the nutrient contents. Both

of our model equations are non-linear reaction-diffusion partial differential equations

(PDEs). Simulations of these models produce large data sets that are difficult to

analyze and interpret. We used a reduced-order modeling technique to interpret

the simulation data in terms of an underlying low-dimensional dynamical system.

The reaction-diffusion equations of our model can be viewed as a stiff system of

equations and require an A-stable time-stepping method. We implement high-order

Implicit Runge–Kutta (IRK) methods for our model PDEs. Although IRK methods

offer A-stability and high-order accuracy, these methods are not widely used in PDE

discretization due to the resulting complicated linear system. To solve these sys-

tems, we have developed a new preconditioner based on a block LDU factorization

with algebraic multigrid subsolves for scalability. We demonstrate the effectiveness

of our preconditioner on two test problems: a heat equation and a double-glazing

advection-diffusion problem, and find that the new preconditioner outperforms the

others currently in the literature.

viii


LIST OF TABLES

2.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Equilibrium Q values for varying total phosphorus Pt and light levels

K0. The range of Q is given when the system exhibits a limit cycle.Values are in bold for Q > Q∗. . . . . . . . . . . . . . . . . . . . . . 20

2.3 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 Common IRK Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Condition numbers of right-preconditioned matrices with precondition-

ers P−1J , P−1

GSL, P−1DU , and P−1

LD applied to a 2D heat equation with

s-stage Radau IIA methods. Here, hx = 2−3 and ht = hp+12s−1x , where

p = 2 is the degree of the Lagrange polynomial basis functions in space.Preconditioners are constructed exactly. . . . . . . . . . . . . . . . . 80

4.3 Condition numbers of left-preconditioned and right-preconditioned ma-trices with preconditioners P−1

GSL and P−1LD applied to a 2D double-

glazing problem with ε = 0.005, and with s-stage Radau IIA meth-

ods. Here, hx = 2−4 and ht = hp+12s−1x , where p = 1 is the degree of

the Lagrange polynomial basis functions in space. Preconditioners areconstructed exactly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Condition numbers of the left-preconditioned system with precondi-tioners PGSL (PGSL with optimal coefficients) and PLD for variousIRK methods applied to 1D and 2D heat problems. Here, hx = 2−3

and ht = hp+1q

x , where p = 2 is the degree of the Lagrange polynomialin space and q is the order of the corresponding IRK method (that is,q = 2s− 1 for Radau IIA and q = 2s− 2 for Lobatto IIIC). Precondi-tioners are constructed exactly. . . . . . . . . . . . . . . . . . . . . . 85

4.5 Iteration counts and elapsed time (times in seconds are shown in paren-theses) for right-preconditioned GMRES to converge with relative resid-ual tolerance 1.0 × 10−8 for a 2D heat problem with s-stage RadauIIA methods with preconditioners PJ , PGSL, PDU , and PLD. Here we

choose ht = hp+12s−1x , where p = 2 is the degree of the Lagrange poly-

nomial in space. Preconditioners are approximated using one AMGV-cycle for each subsolve. . . . . . . . . . . . . . . . . . . . . . . . . 91

ix


4.6 Iteration counts and elapsed time (times in seconds are shown in paren-theses) for right-preconditioned GMRES to converge with relative resid-ual tolerance 1.0 × 10−8 for a 2D heat problem with s-stage LobattoIIIC methods with preconditioners PJ , PGSL, PDU , and PLD. Here we

choose ht = hp+12s−2x , where p = 2 is the degree of the Lagrange poly-

nomial in space. Preconditioners are approximated using one AMGV-cycle for each subsolve. . . . . . . . . . . . . . . . . . . . . . . . . 92

4.7 Iteration counts and elapsed time (times in seconds are shown in paren-theses) for left-preconditioned and right-preconditioned GMRES toconverge with preconditioned relative residual tolerance 1.0× 10−8 fora 2D heat problem with s-stage Radau IIA methods with precondi-tioners PGSL and PLD. Here we keep h−1

x = 128 fixed and vary ht from0.05 to 5.0. Preconditioners are approximated using one AMG V-cyclefor each subsolve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.8 Iteration counts and elapsed time (times in seconds are shown in paren-theses) for left-preconditioned GMRES to converge with preconditionedrelative residual tolerance 1.0 × 10−8 for a 2D heat problem with s-stage Radau IIA methods with preconditioners PJ , PGSL, PGSL, PDU ,

and PLD. Here we choose ht = hp+12s−1x , where p = 2 is the degree of

the Lagrange polynomial in space. Preconditioners are approximatedusing one AMG V-cycle for each subsolve. . . . . . . . . . . . . . . . 94

4.9 Iteration counts and elapsed time (times in seconds are shown in paren-theses) for left-preconditioned GMRES to converge with preconditionedrelative residual tolerance 1.0×10−8 for a 2D heat problem with s-stageLobatto IIIC methods with preconditioners PJ , PGSL, PGSL, PDU , and

PLD. Here we choose ht = hp+12s−2x , where p = 2 is the degree of the La-

grange polynomial in space. Preconditioners are approximated usingone AMG V-cycle for each subsolve. . . . . . . . . . . . . . . . . . . . 95

4.10 Relative error for left-preconditioned GMRES converging to a precon-ditioned relative residual tolerance of 1.0×10−8 for a 2D heat problemwith various IRK methods with preconditioners PJ , PGSL, PGSL, PDU ,

and PLD. Here, hx = 2−7 and ht = hp+1q

x , where p = 2 is the degree ofthe Lagrange polynomial in space and q is the order of the correspond-ing IRK method (that is, q = 2s − 1 for Radau IIA and q = 2s − 2for Lobatto IIIC). Preconditioners are approximated using one AMGV-cycle for each subsolve. . . . . . . . . . . . . . . . . . . . . . . . . 96

x


4.11 Relative error for right-preconditioned GMRES converging to a rela-tive residual tolerance of 1.0×10−8 for a 2D heat problem with variousIRK methods with preconditioners PJ , PGSL, PDU , and PLD. Here,

hx = 2−7 and ht = hp+1q

x , where p = 2 is the degree of the Lagrange poly-nomial in space and q is the order of the corresponding IRK method(that is, q = 2s − 1 for Radau IIA and q = 2s − 2 for Lobatto IIIC).Preconditioners are approximated using one AMG V-cycle for eachsubsolve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.12 Iteration counts and elapsed time (times in seconds are shown in paren-theses) for left-preconditioned GMRES to converge with preconditionedrelative residual tolerance 1.0 × 10−8 for the double-glazing problemwith s-stage Radau IIA methods with preconditioners PGSL and PLD.

Here we choose ht = hp+12s−1x , where p = 1 is the degree of the La-

grange polynomial in space. Preconditioners are approximated usingone AMG V-cycle for each subsolve. . . . . . . . . . . . . . . . . . . . 99

4.13 Iteration counts and elapsed time (times in seconds are shown in paren-theses) for left-preconditioned GMRES to converge with preconditionedrelative residual tolerance 1.0× 10−8 for a 2D double-glazing problemwith s-stage Radau IIA methods with preconditioners PGSL and PLD.Here we keep h−1

x = 128 fixed and vary ht from 0.05 to 5.0. Precondi-tioners are approximated using one AMG V-cycle for each subsolve. . 100

xi


LIST OF FIGURES

2.1 Numerical simulations of Model 2.1 for varying values of total phospho-rus Pt and light levels K0. Solid is algae density u(t), dotted is Daphniagenotype v1, and dashed is Daphnia genotype v2. Genotype v1 is se-lected for in simulations boxed in red and genotype v2 is selected for insimulations boxed in blue. Persistence of both genotypes is seen whenK0 = 1, Pt = 0.025 and K0 = 2, Pt = 0.04, boxed in green. Both geno-types die off in high light, low P conditions K0 = 3, Pt = 0.025, 0.03. 18

2.2 Fitness of the two Daphnia genotypes, equation (2.10) for various foodquantity (u) and quality (Q) conditions under spatially homogenousassumptions and fixed algae population density. . . . . . . . . . . . . 20

2.3 Comparing fitness of the two Daphnia genotypes for various food qual-ity conditions under spatially homogenous assumptions and fixed algaepopulation density. Q∗ = 0.01852 mgP/ mgC divides the space intotwo regions. When Q < Q∗ genotype v1 has a larger fitness than v2.When Q > Q∗, genotype v2 has a larger fitness. . . . . . . . . . . . . 21

2.4 Simulated algal P:C ratios for environmental conditions that resultsin Q (solid black) solutions of Model (2.1) oscillated around Q∗ (dot-ted red). Persistence of both genotypes is seen in cases (a) and (b).Genotype v2 is selected in case (c). . . . . . . . . . . . . . . . . . . . 22

2.5 Time snapshots of simulated population versus depth for four values oftotal phosphorous density Pt. Depth is on the horizontal axis. Algaedensity u is shown in black; densities of Daphnia genotypes v1 andv2 are shown in red and blue, respectively. Normalized food qualityQ/Q∗ is shown in green (Q∗ is defined in equation (2.11)). The scalefor Q/Q∗ is on the right-hand vertical axes. . . . . . . . . . . . . . . 23

2.6 Steady-state solution of simulation with K0 = 2, Pt = 0.025, andκ0 = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Steady-state solution of simulation with K0 = 3, Pt = 0.03, and κ0 = 0.1. 242.8 Numerical simulations of the producer population density for Model

(2.13) for K0 = 0.5 mg C/L (a) and K0 = 1.0 mg C/L (b) and inter-mediate Pt = 0.6 mg P. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xii


2.9 Numerical simulation snap shots for a fixed time showing steady-statebehavior for Model (2.13) for intermediate Pt = 0.6 mgP (a),(c),(e) andhigh Pt = 1.0 mgP (b),(d),(f). The surface light levels is also varied:K0 = 0.5 mg C/L (a)-(b), K0 = 1.0 mg C/L (c)-(d), and K0 = 2.0mg C/L (e)-(f). The horizontal axis is depth in meters, so highestlight levels occur on the left at the surface. Regions where solutionsexhibit sustained oscillations are shaded in gray. Unshaded regionsdepict equilibrium solutions. . . . . . . . . . . . . . . . . . . . . . . 39

2.10 Numerical simulations snap shots for a fixed time showing steady-statebehavior for Model (2.12) (a) and Model (2.13) (b) for K0 = 1.5 mgC/L and P = 0.03 mg P/L. Regions where solutions exhibit sustainedoscillations are shaded in gray. Unshaded regions depict equilibriumsolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.11 Numerical simulations snap shots for a fixed time showing steady-statebehavior for Model (2.12) (a) and Model (2.13) (b) for K0 = 2.5 mgC/L and P = 0.035 mg P/L. Regions where solutions exhibit sustainedoscillations are shaded in gray. Unshaded regions depict equilibriumsolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Singular values of the snapshot matrix W of Model 2.13. The x-axis represents the index ν of the singular value σν , and the y-axisis log (σν/σ1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2 First five left singular vectors of the snapshot matrix W . The x-axisrepresents the depth and the y-axis represents the population in arbi-trary units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Projections of the simulations of model (2.13) onto the reduced basisfor K0 = 0.75 (a) and K0 = 1.5 (b). The x-axis represents time t (indays) and the y-axis represents the population in arbitrary units. . . 60

3.4 Phase Plane of the coefficients of the projections onto the reducedbasis (φ1(z) − φ3(z)) for K0 = 0.75 (a) and K0 = 1.5 (b). The x-axis represents the coefficients of the projections onto φ2(z), the y-axisrepresents the coefficients of the projections onto φ3(z), and the z-axisrepresents the coefficients of the projections onto φ1(z). . . . . . . . 60

3.5 Bifurcation diagram of the model (2.13) for surface irradiance thatcorresponds to surface carrying capacity K0 varying from 0.2 to 2.5.The x-axis represents K0 and the y-axis represents the coefficients ofthe projection onto φ1(z) (a), φ2(z) (b), and φ3(z) (c). . . . . . . . . 61

4.1 Solution plots of (4.2) with Trapezoidal method (a) and BackwardEuler method (b). Figure taken from [17]. . . . . . . . . . . . . . . . 67

xiii


4.2 Eigenvalues of the matrices A, AP−1J , AP−1

GSL, AP−1DU , and AP−1

LD for2D heat problem with Radau IIA s = 2 (a), s = 3 (b), s = 4 (c), ands = 5 (d). The x-axis is the real part and the y-axis is the imaginarypart of the eigenvalue. Preconditioners are constructed exactly. . . . . 83

4.3 Eigenvalues of the matrices A, P−1GSLA, and P−1

LDA for 2D double-glazing problem with ε = 0.005, and with Radau IIA s = 2 (a), s = 3(b), s = 4 (c), and s = 5 (d). The x-axis is the real part and they-axis is the imaginary part of the eigenvalue. Preconditioners areconstructed exactly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Eigenvalues of the matrix A, AP−1GSL, and AP−1

LD for 2D heat problemwith Radau IIA s = 2 (a), s = 3 (b), s = 4 (c), and s = 5 (d).The x-axis is the real part and the y-axis is the imaginary part of theeigenvalue. Preconditioners are constructed exactly. . . . . . . . . . . 86

4.5 Eigenvalues of the matrix A, AP−1GSL, and AP−1

LD for 2D heat problemwith Lobatto IIIC stages s = 3 (a) and s = 4 (b). The x-axis isthe real part and the y-axis is the imaginary part of the eigenvalue.Preconditioners are constructed exactly. . . . . . . . . . . . . . . . . 87

xiv


CHAPTER 1

INTRODUCTION

Important progress in understanding ecological and evolutionary dynamics has

been made through the development of the theory of ecological stoichiometry [55].

This theory considers the balance of multiple chemical elements and how the relative

abundance of essential elements, such as carbon (C), nitrogen (N), and phosphorus

(P), in organisms affects ecological dynamics. Empirical efforts and models devel-

oped under this theory have advanced our understanding of ecological interactions

[3, 24]. The wide variety of stoichiometric food web models that have been proposed

and analyzed incorporate the effects of both food quantity and food quality into a

single framework that produces rich dynamics [2, 35, 20, 58, 59, 45, 44, 63]. Stoichio-

metric models allow one to investigate the effects of nutrient stressors on population

dynamics and track the trophic transfer of energy and nutrients [43].

Light availability plays an important role in the dynamics of an aquatic ecosys-

tem [26, 62, 27]. Irradiance is not uniform throughout a lake; it depends on depth.

Cantrell et al. [9] provides evidence that the spatial location effects population dy-

namics of an ecological interaction. Therefore, to describe the spatial variation of light

irradiance together with the changes of population densities with respect to spatial

location and time, we need a partial differential equation (PDE) model. Movement of

biological species is a random process which can be modeled by diffusion. Diffusion

describes mass transport from high to low concentration. Diffusion together with

a reaction term called a reaction-diffusion model provides a framework for studying

population dynamics of interacting species. A detailed formulation and analysis of

1


reaction-diffusion equations for spatial ecology can be found in [7]. In Chapter 2,

we develop producer-grazer ecosystem models that consider both stoichiometry and

spatial heterogeneity.

Abrams [1] noted that one of the current gaps in the theory of evolutionary

predator-prey interactions involves characterizing the general importance of spatial

heterogeneity on evolutionary dynamics. Cantrell et al. [9] identified the interplay

of spatial dynamics and evolutionary theory as an emerging challenge in spatial ecol-

ogy. While several advances have been made in understanding how spatial dynamics

influence evolutionary theory [22, 25, 8], the connections between genotypic selection

in spatially heterogeneous environments and stoichiometry is not fully understood.

In Section 2.1, we develop a spatially heterogeneous stoichiometric producer-grazer

model with two grazing genotypes to understand genotypic selection process.

In stoichiometric models of producer-grazer interactions, it is important to model

explicitly both food quantity and food quality since they determine the growth rate

of consumers as well as their odds of survival and extinction [58]. Known stoichiomet-

ric models to date of a two species producer-grazer ecosystem have either neglected

spatial dynamics or failed to track free phosphorus in the media. A spatially heteroge-

neous stoichiometric model of a two species producer-grazer ecosystem was developed

by Dissanayake 2016 [13]. This model tracks both, quantity and quality, of the pro-

ducer population in space, assessing quality through the phosphorus to carbon (P:C)

ratio of the producer. Like the LKE model developed by Loladze et al. 2000 [35]

that it extends, this model neglects the free phosphorus in the media through the

assumption that all phosphorus in the system is kept between the producer and the

grazer. Wang et al. 2008 [58] eliminate this assumption to extend the LKE to a model

2


that tracks phosphorus content in the producer and free phosphorus in the media,

but as the LKE, it neglects spatial dynamics. In Section 2.2 we extend the spatially

heterogeneous model of Dissanayake 2016, to track explicitly phosphorus content in

the producer and free phosphorus in the media.

Simulations of spatiotemporal behavior of biological systems produce large data

sets that can be difficult to analyze and interpret. Reduced-Order Modeling (ROM)

techniques can be used to interpret simulations of a stoichiometric producer-grazer

system in terms of an underlying low-dimensional dynamical system. A well-known

property of the Singular-Value Decomposition (SVD) is that it can produce optimal

low-rank approximations to a matrix [57]. This idea can be generalized to find low-

dimensional models that approximate the behavior of a dynamical system with many

degrees of freedom. This technique is known as reduced-order modeling. The purpose

of this approach is not to reduce the computational complexity of the system, but to

gain some biological insight of a large number of variables in space. In Chapter 3 we

discuss the ROM technique and its use to analyze the dynamics of the mechanistically

derived spatially heterogeneous producer-grazer model developed in Section 2.2. We

obtain and record a set of ‘snapshot’ results from the numerical simulations of our

model to produce a reduced-order basis. Then we project our current simulation into

this basis and use phase plane and bifurcation analysis to analyze the dynamics of

the system.

The reaction-diffusion PDE model of the spatio-temporal stoichiometric producer-

grazer system developed in Section 2.2 can be viewed as a stiff system of equations

and requires certain stability properties for time-stepping methods. Explicit time

integrators for parabolic PDEs are subject to the restrictive timestep limit ht . h2x,

3


so A-stable integrators are essential. It is well known that there are no A-stable ex-

plicit Runge-Kutta methods. According to the second Dahlquist barrier (discussed

in Section 4.2.3) there are no A-stable explicit linear multistep methods and implicit

multistep methods cannot be A-stable beyond order two. But there exist A-stable and

L-stable Implicit Runge–Kutta (IRK) methods at all orders [21, 60]. IRK methods

offer an appealing combination of stability and high order; however, these methods

are not widely used for PDEs because they lead to large, strongly coupled linear sys-

tems. An s-stage IRK system has s times as many degrees of freedom as the systems

resulting from backward Euler or implicit trapezoidal rule discretization applied to

the same equation set. Order-optimal preconditioners for such systems have been

investigated in a series of papers [39, 53]. In Chapter 4 we introduce a new block

preconditioner for IRK methods, based on an LDU factorization. Solves on individual

blocks are accomplished using a multigrid algorithm. We demonstrate the effective-

ness of this preconditioner on two test problems. The first is a simple heat equation,

and the second is a model advection-diffusion problem known as the double-glazing

problem. We find that our preconditioner is scalable (independent of mesh size) and

yields better timing results than other preconditioners currently in the literature.

This paper is organized as follows. In Chapter 2, we develop aquatic ecosystem

models that incorporate both spatial heterogeneity and ecological stoichiometry. In

Section 2.1, we develop a spatial stoichiometric producer-grazer model with two graz-

ing genotypes to get insight into the genotypic selection processes. A second spatial

stoichiometric producer-grazer model is developed in Section 2.2 that tracks explic-

itly the nutrient content both in the producer and in the environment. Chapter 3

starts with a description of the reduced-order modeling technique. In Section 3.3, we

4


discuss a reduced-order modeling approach in analyzing the dynamics of the spatial

stoichiometric producer-grazer model 2.13 developed in Section 2.2. Chapter 4 intro-

duces a new block preconditioning technique for implicit Runge–Kutta methods for

parabolic PDE problems.

5


CHAPTER 2

SPACE-TIME POPULATION MODEL

2.1 Genotypic Selection Model

2.1.1 Introduction

An understanding of evolutionary dynamics requires a good understanding of

genotypic selection. Natural selection can occur when a genotypic variation arises

that corresponds to a consistent variation in fitness. Environmental conditions, such

as varying light levels and nutrient loads, can exert selection pressures giving an

advantage to a particular genotype. However, these types of environmental conditions

often fluctuate. In an aquatic setting, the level of light changes with depth and is

altered by population densities along the water column. Ecological interactions and

population dynamics can significantly alter the amount of dissolved nutrients, such

as phosphorus, available for uptake. Correctly predicting genotypic selection and

species persistence depends on complex interactions of population dynamics, spatial

heterogeneity, and stoichiometric constraints.

Ecological Stoichiometry offers a conceptual framework to investigate the im-

pact of elemental imbalances on evolutionary predator-prey systems. It focuses on

key physiological structures and functions and their associated elemental demands

while considering evolutionary change primarily from the perspective of individual

fitness [32, 16]. Stoichiometric traits, such as organismal phosphorous to carbon

(P:C) ratios, govern ecological processes of nutrient recycling and therefore selection

of stoichiometric traits can have a major impact on the flows of energy and nutrients.

Furthermore, there is evidence that variations in these ratios are connected to evolved

6


differences in organismal growth rates due to the relationship between growth and

increased allocation to P-rich ribosomal RNA (rRNA)[16]. Weider et al. 2005 [61]

empirically investigate the interactions between two Daphnia genotypes under varying

environmental scenarios. They used two genetically distinct clones of Daphnia pulex

with different rRNA intergenic spacer (IGS) lengths. One clone had an rRNA IGS

with 3,800 base pairs and the second clone had an rRNA IGS with 3,300 base pairs.

This difference in IGS spacer length corresponded to variations in growth-rate related

life history parameters supporting the growth rate hypothesis (GRH) [55]. According

to this hypothesis, organisms with high growth rates have high demands for ribosomal

RNA production, and therefore higher P content. Using batch cultures, as well as

continuous flow microcosms, Wider et al. 2005 [61] observed that competition and se-

lection of these two different clones depends on stoichiometric constraints. The clone

with the longer ISG spacer out-competed the other under high algal P:C conditions,

however the second clone out-competed the first under low algal P:C conditions.

Yamamichi et al. 2015 [63] demonstrated that rapid evolution of a predator’s

stoichiometric trait can destabilize predator-prey dynamics through the development

and analysis of spatially homogeneous models. Their starting point is a one prey

and two competing predator model developed by [36]. Contrary to the commonly

assumed competitive exclusion principle (CPE), Loladge et al. 2004 [36] show that

mass balance laws and stoichiometric variability can explain the coexistence of mul-

tiple predators exploiting a single prey at equilibrium. The CPE states that no

equilibrium is possible when n species exploit fewer than n resources. Loladge et al.

2004 [36] point out that although the vast biodiversity that is observed in nature

clearly contradicts the CPE, this principle is found to be robust to a host of mod-

7


els hypothesizing that consumers and resources are in a predator-prey relationship.

Loladge et al. 2004 [36] construct what appears to be the simplest model needed to

address variable prey stoichiometry and competition among herbivores and show that

two predators exploiting a single prey can coexist at a stable equilibrium, which is

furthermore structurally stable in the sense that slight variations in the value of the

parameters do no destroy it.

Expanding the work done by Loladge et al. 2004 [36] and Yamamichi et al. 2015

[63], we develop a spatially heterogeneous model with two asexual genotypes preying

on a single prey subject to stoichiometric constraints. The two genotypes are charac-

terized by their P:C ratios and their maximum ingestion rates. We investigate how

diffusion processes, depth-dependent light levels, and varying food quality influence

population interactions and ultimately, selection for a single genotype or persistence

of both genotypes. This work has been published in [14].

2.1.2 Model development

We begin with the two grazer and single producer model by Loladge et al. 2004

[36], which was used by Yamamichi et al. 2015 [63] as an asexual genotype model.

Let u denote the biomass density of algae (the producer) and v1, v2 denote biomass

densities of the two genotypes of Daphnia grazers. Here, biomass density has units

8


mg C/l. The spatially homogenous model takes the following form:

du

dt= bu

1− u

minK0,

Pt−θ1v1−θ2v2q

− 2∑

i=1

fi(u)vi (2.1a)

dv1

dt= emin

1,Q

θ1

f1(u)v1 − dv1 (2.1b)

dv2

dt= emin

1,Q

θ2

f2(u)v2 − dv2 (2.1c)

where b is the algal maximum growth rate, K0 is the light level which defines the

algae carrying capacity in terms of carbon, Pt is the total amount of phosphorus in

the system, q is the minimum algal P:C ratio required for growth, θ1, θ2 are the con-

stant P:C ratios of the two Daphnia genotypes, e the Daphnia maximum conversion

efficiency, and d the Daphnia loss rate. The algal P:C ratio is denoted by Q and

defined as

Q =Pt − θ1v1 − θ2v2

u,

where the numerator represents the P available for the algae. We assume the algae

uptake dissolves P quickly enough to assume environmental P loads are negligible

and all available P is in the grazer or in the producer population. The Daphnia are

assumed to follow a Holling type II functional response:

fi(u) =ciu

a+ u, for i=1,2

where a is the half saturation constant for grazer ingestion and ci is the maximum

ingestion rate for the two genotypes, i = 1, 2.

We assume the producer follows stoichiometric logistic growth. The minimum

9


function in equation (2.1a) represents the stoichiometric carrying capacity for the

algae population. The use of the minimum function follows directly from Justin

Leibig’s law of the minimum, which states that an organism’s growth will be limited

by whichever single resource is in lowest supply relative to the organism’s needs [55].

The first input, K0 is the light level (mg C/l), assumed constant in the above model.

The minimum function takes this term when algal growth is limited by C. The second

input in this minimum function is the algal carrying capacity when growth is limited

by P. Below we expand K0 to a depth-dependent function.

The minimum functions in equations (2.1b, c) represent the grazers’ stoichio-

metric growth rates. The first input, 1, is used when grazer growth is limited by C.

The second input, Q/θi is used when grazer growth is limited by P. The two grazing

genotypes are characterized by the constant P:C ratio θi and maximum ingestion

rate ci, following Yamamichi et al. [63]. Here, we assume θ1 > θ2. The growth rate

hypothesis (GRH) predicts that organisms with higher maximum growth rates have

higher P:C ratios due to the increased demand for ribosomal RNA production needed

for growth [55]. Following the GRH, we assume that genotype v1, with a higher P:C

ratio (θ1 > θ2), has a higher growth rate and consider the maximum ingestion rate to

be higher, c1 > c2.

Model 2.1 takes similar assumptions as the base stoichiometric predator-prey

model developed by Loladge et al. [35], as well as an assumption following the GRH.

The major assumptions of the model are listed below:

A1: The total mass of phosphorus in the entire system is fixed, i.e., the system

is closed for phosphorus with a total of Pt (mgP/L).

A2: P:C ratio in the producer varies, but it never falls below a minimum q

10


(mgP/mgC); the grazers maintain constant P:C, θ1 and θ2 (mgP/mgC).

A3: All phosphorus in the system is divided into two pools: phosphorus in the

grazer and phosphorus in the producer.

A4: Following the GRH, higher growth rates correspond to higher P:C ratios;

θ1 > θ2 and c1 > c2.

2.1.2.1 Spatially heterogeneous model

Here, we spatially expand the two asexual grazer genotype model presented in

[63] and incorporate diffusion and depth-dependent light levels. A detailed model of

plankton motion would be extraordinarily complex. Even in still water they swim and

self-regulate buoyancy in response to environmental conditions. In realistic conditions

in a lake, they are carried with macroscopic flow caused by thermal gradients and

wind-driven circulation. Our transport model is greatly simplified. In the present

paper, we consider only vertical motion and ignore buoyancy effects. Swimming is

modeled as a random walk which contributes a diffusion term with coefficient Di for

species i = u, v. Turbulent convection is modeled phenomenologically by an effective

diffusion coefficient Dz [6, 28]. Rather than trying to include boundary layer effects,

we simply take Dz to be constant throughout the domain. Outside of a controlled

laboratory, the turbulent effective diffusivity Dz will normally be larger than the

particle diffusivities Di.

11


Our spatially heterogeneous two grazing genotype model is below:

∂u(x, t)

∂t− ∂

∂z

[(Dz +Du)

∂u

∂z

]= Ru(u, v1, v2, K) (2.2a)

∂v1(x, t)

∂t− ∂

∂z

[(Dz +Dv)

∂v1

∂z

]= Rv1(u, v1, v2) (2.2b)

∂v2(x, t)

∂t− ∂

∂z

[(Dz +Dv)

∂v2

∂z

]= Rv2(u, v1, v2) (2.2c)

where Ru, Rv1 , Rv2 are the ecological dynamics from the right hand side of Model 2.1

with a depth-dependent light level for K0, described below (Section 2.1.2.2). Param-

eter values are listed in Table 2.1.

In our one-dimensional model, the domain is [0, H] with the water’s surface at

z = 0. At both boundaries, no-flux conditions hold for all populations:

− (Dz +Du)∂u

∂z= 0 and − (Dz +Dvi)

∂vi∂z

= 0 (2.3)

for i=1,2.

Initial conditions are assumed constant in space:

u(x, 0) = u0(x), vi(x, 0) = vi0(x) for i = 1, 2. (2.4)

2.1.2.2 Light absorption

Light irradiance I in a one-dimensional absorbing medium obeys the equation of

radiative transfer [51, 34],

dI

dz= −κI (2.5)

12


with boundary condition at the surface

I(0) = I0, (2.6)

where κ is an absorption coefficient. In cases where κ is a constant, the familiar

Lambert-Beer exponential law I(z) = I0 exp(−κz) is recovered. In an aquatic ecosys-

tem, plankton themselves act as absorbers, so the absorption coefficient will depend

on plankton population as

κ(u, v1, v2) = κ0 + σuu+2∑i=1

σvvi, (2.7)

where κ0 is the absorption coefficient for plankton-free water, σu, σv are the absorption

cross sections per unit of carbon in species u and v, respectively. In this preliminary

study we regard the absorptivity κ as constant in space. Light availability is an

important factor for the producer’s carrying capacity, when producer growth is limited

by carbon. The LKE model assumes K0 is positively correlated with light irradiance.

Given a particular irradiance and ample nutrients, the producer density grows but

eventually stabilizes at K0 due to shelf-shading. In our extended model the irradiance

varies with depth following eq. (2.5). Here, we assume a linear relationship between

the irradiance and carbon-dependent producer carrying capacity K,

K(z) = αI(z), (2.8)

where α is a conversion coefficient correlating irradiance with the producer carrying

capacity, under environmental conditions where growth is limited by light-supplied

13


carbon. The largest carrying capacity will occur on the surface where light irradiance

is the largest I0. Non-spatial stoichiometric models typically assume algal carrying

capacity K0 ∈ (0, 3) mg C/L. In order to relate to these models and stay within

similar parameter space we assume the light irradiance at the surface, I0 corresponds

to K(0) = K0 and parameterize α accordingly. Given that global average irradiance

is 1,366 watts/m2 [5], We assume α=1.098 mg C/m/watts and consider values K0 ∈

(0, 3) mg C/L.

In principle, were the mean turbulent flow profile known we could estimate an

effective diffusion coefficient from mixing length theory [6]. Instead, we use emprical

values taken from a survey of the literature. These estimates have been inferred in-

directly from, for example, measurements of temperature profiles [38], measurements

of tracer species, and frequency spectra of buoyant oscillations.

Table 2.1. Model Parametersparameters description value

b algal maximum growth rate 1.2/dayq algal minimum P:C ratio 0.0038 mg P/mg Cθ1 P:C ratio of Daphnia genotype 1 0.03 mg P/mg Cθ2 P:C ratio of Daphnia genotype 2 0.015 mg P/mg Cc1 max ingestion rate of Daphnia genotype 1 0.81/dayc2 max ingestion rate of Daphnia genotype 2 0.5/daye Daphnia maximum conversion efficiency 0.8a half saturation constant of Daphnia ingestion 0.25 mc C/ld Daphnia loss rate 0.25/dayDz Effective diffusivity 0.1 m2/dayDu algal particle diffusivity 0.0001 m2/dayDv Daphnia particle diffusivity 0.01 m2/dayκ0 Constant absorption coefficient 0.1− 0.3m−1

K0 Carrying capacity at surface z = 0 0-3 mgC/lPt total phosphorus 0-0.4 mgP/l

14


2.1.3 Numerical methods

We conduct several numerical experiments of Models 2.1 and 2.2 in order to

investigate genotypic selection under various environmental conditions. We first ex-

plore the dynamics of the spatially homogenous Model 2.1 under a range of light

levels (K = 0.5− 3 mg C/l) and phosphorus loads (Pt = 0.025− 0.04 mg P/l). These

diverse environmental conditions yield a variety of population dynamics including

selection of genotype v1 over genotype v2, selection of genotype v2 over genotype v1,

persistence of both genotypes, and extinction of both. The results of these spatially

homogenous numerical experiments help guide the experiments of the spatially het-

erogenous model. We explore the dynamics Model 2.2 under similar ranges of light

levels and phosphorus loads. In particular, we examine environmental conditions that

yielded persistence of both genotypes in the spatially homogeneous case and vary the

absorption coefficient κ0.

The reaction-advection diffusion equation Model 2.2 together with equation eq.

(2.5) for radiative transport, form a system of nonlinear partial differential equations

which must be solved numerically. Our simulation methods are based on those de-

veloped by [13] for an algae-daphnia ecosystem with stoichiometry and diffusion. We

semidiscretize the equations in space using a Galerkin finite element method; this

method handles the no-flux boundary conditions eq. (2.3) very simply and is eas-

ily generalizable to problems in three dimensions with complex geometry. Galerkin

discretization of an advection-diffusion system is stable provided the element Peclet

number

Peh = maxi|ωi|h/ (Dz +Di)

for i = u, v is less than unity [19, 30]. Our element size h is chosen to respect this

15


condition.

We note that while the authors in [27] formulated the radiative transport com-

ponent of the model as an integral equation, we work directly with the equivalent

differential equation formulation given in eq. (2.5). There are two advantages to this

approach: first, it is simpler to program with standard PDE discretization software

tools such as Long et al. 2010 [37]; second, the equations remain entirely local, with

the effect that the matrices arising in the linear subproblems are sparse rather than

dense.

For time integration, the implicit trapezoidal rule (TR) is used. Since the TR is

implicit, a nonlinear solve is required at every timestep. Because no time derivative

appears in eq. (2.5), the complete system after discretization is actually a differential-

algebraic system [4], with eq. (2.5) acting as an algebraic constraint with respect to

time integration. The TR is suitable for differential-algebraic systems, and as an A-

stable method is not subject to a limiting timestep. With linear Lagrange elements

for spatial discretization and the TR for time integration, the method is second order

accurate in both time and space [13].

The stoichiometric reaction terms involve minimum functions, which are non-

differentiable. To avoid possible numerical difficulties during nonlinear solves, a

smoothed minimum function

smooth min(x, y) =xy

(xp + yp)1p

(2.9)

was used [13]. After experimentation we found that p ≈ 16−32 gave an approximate

minimum function sufficiently sharp to preserve the system’s correct dynamics yet

sufficiently smooth to avoid numerical difficulties.

16


Simulations were implemented using the Sundance PDE simulation toolkit [37],

a component of the Trilinos library for high-performance computation [23]. Sundance

has a built-in automatic differentiation capability for computation of the Jacobian op-

erators needed in nonlinear solves with a Newton-type method. Linear subproblems

were solved using the Amesos sparse direct solver library [50], and nonlinear equa-

tions were solved using a Newton-Armijo method [33]. Extensive validation tests of

the simulation software were carried out for a two-species stoichiometric model with

radiative transport in [13].

2.1.4 Results

The results of our numerical experiments varying environmental conditions of

Models 2.1 and 2.2 are presented below.

2.1.4.1 Spatially homogeneous results

First we consider the spatially homogenous dynamics of Model 2.1. Fig. 2.1

shows how different environmental conditions affect the selection process under spa-

tially homogenous assumptions. Here, simulations are boxed in red when genotype

v1 is selected, blue when genotype v2 is selected, and green when both genotypes

persists.

Under low light level K0 = 0.5 mg C/l (Fig. 2.1 top row) genotype v1 is selected

for and genotype u2 dies out regardless of the level of Pt. In this case there is a

stable boundary equilibrium E(1) =(u(1), v

(1)1 , 0

). These low light conditions result

in high food quality, Q, and genotype v1 has the advantage. As light levels increase

oscillations emerge as the system exhibits a Hopf bifurcation (Fig. 2.1 second row).

Environmental conditions K0 = 1 mg C/l and Pt = 0.025 mg P/l allow the persistence

17


Figure 2.1. Numerical simulations of Model 2.1 for varying values of total phosphorusPt and light levels K0. Solid is algae density u(t), dotted is Daphnia genotype v1, anddashed is Daphnia genotype v2. Genotype v1 is selected for in simulations boxed inred and genotype v2 is selected for in simulations boxed in blue. Persistence of bothgenotypes is seen when K0 = 1, Pt = 0.025 and K0 = 2, Pt = 0.04, boxed in green.Both genotypes die off in high light, low P conditions K0 = 3, Pt = 0.025, 0.03.

of both genotypes in a stable periodic solution or limit cycle (green box). For K0 = 1

mg C/l increasing Pt shortens the period of the limit cycle and causes genotype v1

to die out. As K continues to increase the cycles collapse (Fig. 2.1 rows 3, 4) and

another stable equilibrium emerges, E(2) =(u(2), 0, v

(1)2

). This change in selection

is due to the stoichiometric constraints incorporated into the model. High values of

K result in low Q, food quality. Under lower food quality conditions genotype v2

has the advantage and is selected for. Environmental conditions K0 = 2 mg C/l and

18


Pt = 0.04 mg P/l provide another scenario for persistence of both genotypes (green

box). Under very high light levels K0 = 3 mg C/l neither genotype can survive if P

levels are too low (Fig. 2.1 row 4). Here, food quality has become too low to support

the grazers. As P levels increase, food quality increases enough to allow persistence

of genotype v2.

Under spatially homogenous conditions, we investigate the fitness of the two

genotypes assuming algal population density is fixed at an equilibrium value. While

a good approximation of fitness is based on dominant Lyapunov exponents [41], in

an equilibrium environment, fitness can be measured by the per capita growth rate

of the clone [40]. Here, we approximate fitness at equilibria values. Let Fv1 and Fv2

denote the fitness of the two genotypes, where

Fv1 =dv1/dt

v1

= emin

1,Q

θ1

c1u

a+ u− d (2.10a)

Fv2 =dv2/dt

v2

= emin

1,Q

θ2

c2u

a+ u− d. (2.10b)

The fitness functions are plotted for varying values of food quantity, u and quality Q

in Fig. 2.2.

At equilibrium conditions, there exists a threshold algal P:C ratio where the

fitnesses of the two genotypes are equal. We denote this food quality threshold as Q∗

where

Q∗ =c2

c1

θ1 = 0.01852, (2.11)

given parameter values in Table 2.1. Genotype v1 has an advantage when Q > Q∗,

whereas genotype v2 has an advantage when Q < Q∗ (Fig. 2.2, 2.3). Table 2.2

shows a summary of Q values corresponding to the simulations presented in Fig.

19


Figure 2.2. Fitness of the two Daphnia genotypes, equation (2.10) for various foodquantity (u) and quality (Q) conditions under spatially homogenous assumptions andfixed algae population density.

2.1, where values are bold for Q > Q∗. Figure 2.4 shows Q for the cases where

Q oscillates around Q∗. Persistence of both genotype can occur when the systems

exhibits periodic solutions such that Q oscillations around Q∗ (Fig. 2.4ab). Although

Q oscillates around Q∗ when K0 = 2 mg C/l and Pt = 0.035 mg P/l persistence of

both genotypes does not occur (Fig. 2.4c). Here Q does not spend enough time above

Q∗ to allow co-persistence and genotype v1 is selected.

Table 2.2. Equilibrium Q values for varying total phosphorus Pt and light levels K0.The range of Q is given when the system exhibits a limit cycle. Values are in boldfor Q > Q∗.

Pt = 0.025 Pt = 0.03 Pt = 0.035 Pt = 0.04

K0 = 0.5 0.082 0.1120 0.1439 0.1757K0 = 1 0.0165-0.257 0.0205-1.54 0.025-7.94 0.0315-30K0 = 2 0.0106 0.0109 0.0114-8.3 0.0132-125K0 = 3 0.008 0.009 0.0102 0.0103

20


0.040 0.005 0.01 0.02 0.025 0.035

Q (mgC/mgP)θ2

Fv1 < Fv2 Fv1 > Fv2

θ1

Q*

Figure 2.3. Comparing fitness of the two Daphnia genotypes for various food qualityconditions under spatially homogenous assumptions and fixed algae population den-sity. Q∗ = 0.01852 mgP/ mgC divides the space into two regions. When Q < Q∗

genotype v1 has a larger fitness than v2. When Q > Q∗, genotype v2 has a largerfitness.

2.1.4.2 Spatially heterogeneous results

We conducted a series of numerical experiments of Model 2.2 in which phospho-

rous supply Pt was varied while K0 = 2 and κ0 = 0.1 were held constant. In most of

the spatial domain these solutions oscillate in time. In Figure 2.5 snapshots in time

of u, v1, v2 and Q/Q∗ versus depth are shown. The snapshot time t = 1000 days was

arbitrarily chosen. Though these snapshots cannot capture the oscillatory dynamics

of the system, they do represent well the behavior we wish to highlight here: the

selection of subpopulations v1 and v2 at different depths.

In Figures 2.5 (a), (b), and (c), corresponding to Pt = 0.025, 0.03, and 0.035,

we see co-persistence of v1 and v2 at shallow depths but disappearance of v2 from

deeper water where less light is available. In (a) and (b) v2 apparently outcompetes

v1 at shallow depths. In (c) v1 dominates throughout the spatial domain but does not

drive v2 to extinction; a small v2 population persists near the surface. In the higher

phosphorous level shown in (d), v2 cannot survive.

21


5000 100 200 300 400

0.01

10-1

time (days)

mg

P/ m

g C

Q*

100

101

102

(a) K0 = 1, Pt = 0.025

5000 100 200 300 400

time (days)

mg

P/ m

g C

10-2

10-1

Q*

100

101

102

(b) K0 = 2, Pt = 0.04

5000 100 200 300 400

time (days)

mg

P/ m

g C

10-2

10-1

Q*

100

101

102

(c) K0 = 2, Pt = 0.035

Figure 2.4. Simulated algal P:C ratios for environmental conditions that results in Q(solid black) solutions of Model (2.1) oscillated around Q∗ (dotted red). Persistenceof both genotypes is seen in cases (a) and (b). Genotype v2 is selected in case (c).

In Figures 2.5 (a) and (b) it is interesting to notice that the point where v1 = v2

is at very nearly the same depth at which Q/Q∗ = 1. As described in the argument

leading up to equation (2.11), this is where both genotypes are equally fit. However,

in (c) we see dominance of v1 even at those depths where Q < Q∗. Finally, in (d)

where Q Q∗ throughout the spatial domain, v2 has gone extinct.

All solutions shown in Figure 2.5 exhibited persistent oscillations in time; how-

ever, for other parameter values we did find solutions that reached a stable steady

state. In Figure 2.6 we show the steady state solution of a model with K0 = 2,

Pt = 0.025, and κ0 = 0.2. This experiment has the same phosphorous concentra-

tion and irradiance as the experiment shown in Figure 2.5(a), but absorptivity has

been increased from κ0 = 0.1 to 0.2. Unlike the spatially homogeneous model, here

we found co-persistence of both genotypes at equilibrium. Model 2.1 only exhibits

co-persistence during limit cycles (Figure 2.1).

An additional numerical experiment is shown in Figure 2.7 with very high light

level at the surface K0 = 3 mg C/l and Pt = 0.03 mg P/l. Here, we see selection

of genotype v2 near the surface, where food quality is low. Both genotypes persist

22


0 5 10 15 20

Depth (m)

0.0

0.5

1.0

1.5

2.0

Densi

ty (

mg C

/L)

=0.1 Pt =0.025 K0 =2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Q/Q∗

AlgaeDaphnia 1Daphnia 2

Q/Q ∗

(a) Pt = 0.025

0 5 10 15 20

Depth (m)

0.0

0.5

1.0

1.5

2.0

Densi

ty (

mg C

/L)

=0.1 Pt =0.03 K0 =2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Q/Q∗


Q/Q ∗

(b) Pt = 0.03

0 5 10 15 20

Depth (m)

0.0

0.5

1.0

1.5

2.0

Densi

ty (

mg C

/L)

=0.1 Pt =0.035 K0 =2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Q/Q∗


Q/Q ∗

(c) Pt = 0.035

0 5 10 15 20

Depth (m)

0.0

0.5

1.0

1.5

2.0

Densi

ty (

mg C

/L)

=0.1 Pt =0.04 K0 =2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Q/Q∗


Q/Q ∗

(d) Pt = 0.04

Figure 2.5. Time snapshots of simulated population versus depth for four valuesof total phosphorous density Pt. Depth is on the horizontal axis. Algae density uis shown in black; densities of Daphnia genotypes v1 and v2 are shown in red andblue, respectively. Normalized food quality Q/Q∗ is shown in green (Q∗ is defined inequation (2.11)). The scale for Q/Q∗ is on the right-hand vertical axes.

at intermediate depths and v1 is selected at low depths, where food quality remains

high.

2.1.5 Discussion

This study explores the interesting interplay between genotypic selection subject

to stoichiometric and spatial constraints. We conducted numerical experiments of

both spatially homogeneous and heterogeneous environments that investigate producer-

23


0 5 10 15 20

Depth (m)

0.0

0.5

1.0

1.5

2.0

Densi

ty (

mg C

/L)

=0.2 Pt =0.025 K0 =2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Q/Q

∗


Q/Q ∗

Figure 2.6. Steady-state solution of simulation with K0 = 2, Pt = 0.025, and κ0 = 0.2.

0 5 10 15 20

Depth (m)

0.0

0.5

1.0

1.5

2.0

Densi

ty (

mg C

/L)

=0.1 Pt =0.03 K0 =3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Q/Q

∗


Q/Q ∗

Figure 2.7. Steady-state solution of simulation with K0 = 3, Pt = 0.03, and κ0 = 0.1.

grazer population dynamics, as well as selection and co-persistence of grazer genotypes

for various environmental light and nutritional conditions. Simulations of the spa-

tially homogeneous model (2.1) show that diverse population dynamics are possible

including selection of a single genotype, persistence of both genotypes during oscilla-

tory solutions, and complete extinction of the grazer. Similar to empirical evidence

observed by Weider at al. [61], we identified environmental conditions for each of the

genotypic clones to out-compete the other, as well as clonal coexistence. Compar-

24


ing the fitness of the two genotypes revealed an important stoichiometric threshold

of food quality (Q∗), useful in predicting genotypic selection. Co-persistence is only

observed when solutions are on a limit cycle where food quality oscillates around this

threshold.

Numerical experiments of the spatially heterogeneous model (2.2) yield richer

dynamics. Incorporating diffusion and depth-dependent light levels allows selection

and persistence of both genotypes to vary across depth. We observed scenarios of

different genotypic selection for different depth. We also observed interesting cases

of co-persistence of both genotypes at the same depth at limit cycles, as well as at

equilibria.

In this study, we considered two grazer genotypes with different stoichiometric

ratios θ1, θ2 and maximum growth rates c1, c2 following the GRH. Similar to the asex-

ual clonal model approach in [63], our spatially homogeneous model is mathematically

identical to the stoichiometric model of one single producer and two competing graz-

ers developed in [36]. In addition we investigate fitness of the two genotypes assuming

that algal density is fixed at an equilibrium value. We observe that at equilibrium

conditions, there exists a threshold algal P:C ratio where the fitnesses of the two

genotypes are equal and we identify the escenarios where each genotype prevails as

well as persistence of both genotypes relative to this threshold. In our second model,

we spatially expand the two asexual grazer genotype model to include diffusion and

depth-dependent light levels. In future work it would be interesting to consider the

quantitative genetic approach presented in [63] and allow the adaptive trait to vary

according to the fitness gradient.

The above models assume that all phosphorus in the systems is located inside

25


the organisms and does not account for free nutrients in the environment. This

assumption is based on the fact that algae take up nutrient quickly and P only spends

a short time in the environment. However, when nutrient pools in the environment

are relevant to the dynamics, free nutrients must be tracked in the model. This

has been done, for example, in the extended models in [63, 46]. We will include

tracking of free phosphorus in future expansions of our model. Temperature is another

important aspect that may influence spatial ecological dynamics that is worth further

investigation. In fact, the authors in [42] empirically looked at different temperatures

and found evidence that suggests the GRH does not apply at different temperatures.

In this paper we have kept the particle transport and radiative transfer models

very simple so that we could concentrate on effects primarily due to stoichiometry.

Future enhancements to the transport model will include buoyancy. Future enhance-

ments to the radiation model will include concentration-dependent absorptivity and

also diurnal and seasonal variations in surface irradiance. Another avenue for ex-

ploration is the inclusion of variation of phosphorous concentration with space and

time.

Including buoyancy and concentration-dependent absorptivity with stoichiome-

try in an evolutionary context is interesting because plankton can self-regulate buoy-

ancy to seek food or light; this adds the issue of evolutionary strategy and will best

be simulated through adaptive dynamics.

The most challenging aspect of enriching our model will be drawing useful bio-

logical conclusions from spatiotemporally variable results in a large multidimensional

parameter space. In this paper, we have investigated the applicability of the Q∗ fit-

ness balance criterion – developed from a space-independent model – to a spatially

26


variable model, and we have found that it can often predict the spatial regions in

which different genotypes will predominate. However, as discussed above, the Q∗

criterion is not uniformly successful. Further investigation of why the Q∗ criterion

sometimes fails is necessary. We will explore dimension reduction techniques to seek

low-dimensional dynamics within this very complex system.

2.2 Space-Time Mechanistic Model

2.2.1 Introduction

In this section, we study the dynamics of a two species producer-grazer (algae-

Daphnia) ecosystem when the essential nutrient phosphorus is tracked in both the

producer and the media. Our main references are the LKE model developed by Lo-

ladze et al. 2000 [35], its spatial extension by Dissanayake 2016 [13] and an extension

by Wang et al. 2008 [58]. The LKE is a two dimensional stoichiometric model where

carbon is used to measure the biomass of the populations, and phosphorus is implicitly

tracked through the phosphorus to carbon (P:C) ratio. The main model assumptions

are:

A1: The total mass of phosphorus in the entire system is fixed, i.e., the system

is closed for phosphorus with a total of P (mgP/L).

A2: P:C ratio in the producer varies, but it never falls below a minimum q

(mgP/mgC); the grazers maintain constant P:C, θ (mgP/mgC).

A3: All phosphorus in the system is divided into two pools: phosphorus in the

grazer and phosphorus in the producer.

The incorporation of chemical heterogeneity and stoichiometric constraints in the

LKE model leads to complex dynamics with multiple equilibria, where bistability

27


and deterministic extinction of the grazer are possible. This model highlights the

important fact that energy enrichment of producer-grazer systems is dynamically

different than nutrient enrichment.

Dissanayake 2016 [13] extended the LKE model to include spatial dynamics. The

extended model still uses the three assumptions above and deals with the phosphorus

in the same way as the LKE, only tracking it implicitly through the (P:C) ratios of

the producer and consumer. The assumption A3 is also dropped in the spatially ho-

mogeneous model developed by Wang et al. [58], which extends the LKE to explicitly

track phosphorus in the producer and the media. Such considerations result in a 4-D

model which in their case leaves out consideration of spatial dynamics of the system.

With diffusivity not present in their model, the sole verification that the total amount

of phosphorus in the system is constant allows them to drop one equation and work

with a reduced 3-D model.

Our starting point is the spatial extension of the LKE model proposed by Dis-

sanayake 2016 [13]. Then, similarly to Wang et al [58], we arrive at a 4-D model

mechanistically formulated to explicitly track phosphorus in the producer and the

media. Since our model incorporates spatial dynamics, unlike [58], we do not reduce

it down to a 3-D model. To make that work in a spatially heterogeneous model,

we would need to add the rather unrealistic assumption that the diffusivities of all

variables in the system are equal. In reality, Daphnia diffusivity is higher than the

rest, and we choose to stay in the 4-D model.

Numerical simulations of our model reveal rich dynamics where the existence

and stability of equilibria and limit cycles depend on depth. For model comparisons,

we ran numerical simulations of a modified version of the model proposed by Dis-

28


sanayake 2016 [13]. This allowed us to compare a spatial model that uses the above

three assumptions with our model that drops assumption A3 and explicitly tracks

free P in space. Under the same parameter sets, we observed qualitative differences

between these two models, which highlight the importance of considering environ-

mental nutrients loads in stoichiometric models. This work has been published in

[46].

2.2.2 Model development

We begin with the spatially heterogeneous model developed in [13], where as-

sumptions A1-A3 from [35] are kept. Let u be the biomass density of producer

(algae) and v be the biomass density of grazer (Daphnia). The model is given by:

∂u(x, t)

∂t− ∂

∂z

[(Dz +Du)

∂u

∂z

]= bu

1− u

minK(z), Pt−θv

q

− f(u)v (2.12a)

∂v(x, t)

∂t− ∂

∂z

[(Dz +Dv)

∂v

∂z

]= e min

1,Q

θ

f(u)v − d v, (2.12b)

where Dz is the effective turbulent diffusivity; Du, the algal particle diffusivity; Dv,

the Daphnia particle diffusivity; b, the algal maximum growth rate; Pt, the fixed

total amount of phosphorus in the system; q, the minimum algal P:C ratio required for

growth; θ, the constant P:C ratio of the Daphnia; e, the Daphnia maximum conversion

efficiency and d, the Daphnia loss rate. The spatial coordinate is represented by

x = (x, y, z), although in our analysis below we assume a one dimensional model that

represents a water column with z as depth. The algal P:C ratio is denoted by Q and

is defined as

Q =Pt − θ v

u,

29


where the quantity Pt − θv represents the available phosphorus for the algae. Note

that this follows from assumption A3 above, as the P in the algae equals the total

amount in the system, Pt, minus the P in the grazer population, θv.

In this model, K(z) represents the algae carrying capacity in terms of carbon,

which is measured through the depth dependent irradiance discussed in Section 2.2.2.1

of this paper. The Daphnia biomass density is assumed to follow a Holling type II

functional response:

f(u) =c u

a+ u,

where a is the half saturation constant for grazer ingestion and c is the maximum

ingestion rate.

Following Wang et al. 2008 [58], where P is tracked in a spatially homogeneous

model, we drop assumption A3 and explicitly track the phosphorus in algae and free

phosphorus in the media. Here, we now assume that all P in the system is divided

into three pools: P in the grazer, P in the producer, and free P in the environment.

Let Pa be the density of phosphorus in algae and Pf be the density of free phosphorus

in the media. These P quantities depend on space and time. Our extended model

30


takes the following form:

∂u(x, t)

∂t− ∂

∂z

[(Dz +Du)

∂u

∂z

]= bu

1− u

minK(z), Pa

q

− f(u)v (2.13a)

∂v(x, t)

∂t− ∂

∂z

[(Dz +Dv)

∂v

∂z

]= e min

1,Q

θ

f(u)v − d v (2.13b)

∂Pa(x, t)

∂t− ∂

∂z

[(Dz +Du)

∂Pa∂z

]= g(Pf )u−

Pauf(u)v − d Pa (2.13c)

∂Pf (x, t)

∂t− ∂

∂z

[(Dz +Dp)

∂Pf∂z

]= −g(Pf )u+ d Pa + θ d v

+

(Pau− emin

θ,Pau

)f(u)v, (2.13d)

where Dp is the particle diffusivity of phosphorus and d the loss rate of phosphorus

for the producer. The algal P:C ratio Q in equation (2.13b) here takes the form

Q =Pau.

The phosphorus uptake rate of the producer is given by g(Pf ) and is also assumed to

follow a Holling type II functional response:

g(Pf ) =c Pfa+ Pf

.

Here a is the phosphorus half saturation constant, and c is the maximum phosphorus

uptake rate of the algae population. The remaining parameters in Model 2.13 mean

the same as in Model 2.12. The parameter values that we use in the numerical

simulations are given in Table 2.3.

Free phosphorous occurs in many forms in our system, including phosphorous

31


ions and phosphorous bound in dead organisms and macromolecular metabolic waste.

Typical diffusion coefficients for atoms, ions, and small molecules in water are on order

10−5 cm2 s−1 [10], which in our units is 8.64×10−5 m2 day−1. We adopt a nominal value

of 10−4 m2 day−1 for Dp; note that this is identical to our choice of algal diffusivity

Du.

The domain of our one-dimensional model is [0, H] with the water’s surface at

z = 0 and H is the total depth of the water column in meters. At both boundaries,

no-flux conditions hold for both populations and P quantities.

− (Dz +Du)∂u

∂z= 0, − (Dz +Dv)

∂v

∂z= 0,

− (Dz +Du)∂Pa∂z

= 0, and − (Dz +Dp)∂Pf∂z

= 0.

(2.14)

Initial conditions are assumed constant in space:

u(x, 0) = u0(x), v(x, 0) = v0(x), Pa(x, 0) = Pa0(x), Pf (x, 0) = Pf0(x). (2.15)

The total amount of phosphorus in the system Pt is the sum of the phosphorus in the

environment, phosphorus in the grazer, and phosphorus in the producer and thus is

given by

Pt (t) =

∫ H

0

(Pf + Pa + θ v) dz. (2.16)

32


Using the boundary condition (2.14), we have

dPtdt

=

∫ H

0

(∂Pf∂t

+∂Pa∂t

+ θ∂v

∂t

)dz

=

∫ H

0

[∂

∂z

(Dz +Dp)

∂Pf∂z

+

∂

∂z

(Dz +Du)

∂Pa∂z

+θ

∂

∂z

(Dz +Dv)

∂v

∂z

]dz

= 0.

Thus the total phosphorus in system (2.13) remains constant at all time. In the special

case where the four variables u, v, Pa and Pf have exactly the same diffusivity, the

global constraint (2.16) implies the local constraint Pt = Pf + Pa + θ v which can be

used to reduce the 4-D Model 2.13 to a 3-D model as in [58]. However, that is not

a realistic assumption since in the real world Daphnia diffusivity is higher than the

diffusivity of the other variables in the system. Here, we consider the 4-D Model 2.13

with the global constraint given by equation (2.16).

2.2.2.1 Light absorption

In a 1-dimensional medium, the irradiance I obeys the equation of radiative

transfer [34, 51],

dI

dz= −κ I (2.17)

with boundary condition at the surface

I(0) = I0, (2.18)

where κ denotes the absorption coefficient and I0 is the irradiance at the surface.

33


In cases where κ is a constant, the familiar Lambert-Beer’s exponential law I(z) =

I0 exp(−κz) is recovered. In an aquatic ecosystem, the light is absorbed by the water

molecules, dissolved organic matter, phytoplankton population and many other light

absorbing substances [27, 34]. Thus, we assume the absorption coefficient depends

on the density of the phytoplankton population as

κ(u) = κu u+ κbg, (2.19)

where κu is the specific light attenuation coefficient of algal biomass and κbg is the

total background turbidity due to nonphytoplankton components.

Irradiance is an important factor for the producer’s carrying capacity, when pro-

ducer growth is limited by carbon. The LKE model assumes K0 is positively corre-

lated with irradiance. Given a particular irradiance and ample nutrients, the producer

density grows but eventually stabilizes at K0 due to shelf-shading. In our extended

model the irradiance varies with depth following eq. (2.17). Here, we assume a linear

relationship between the irradiance and carbon-dependent producer carrying capacity

K,

K(z) = αI(z), (2.20)

where α is a conversion coefficient correlating irradiance with the producer carrying

capacity, under environmental conditions where growth is limited by light-supplied

carbon. The largest carrying capacity will occur on the surface where irradiance

is the largest I0. Non-spatial stoichiometric models typically assume algal carrying

capacity K0 ∈ (0, 3) mg C/L. In order to relate to these models and stay within

similar parameter space we assume the irradiance at the surface, I0 corresponds to

34


K(0) = K0 and parameterize α accordingly. Given that global average irradiance is

1,366 watts/m2 [34], We assume α=1.098 mg C/m/watts and consider values K0 ∈

(0, 3) mg C/L.

Table 2.3. Model Parameters

parameters description value

b algal maximum growth rate 1.2/dayq algal minimum P:C ratio 0.0038 mg P/mg Cθ P:C ratio of Daphnia 0.03 mg P/mg Cc max ingestion rate of Daphnia 0.81/dayc algal max phosphorus uptake rate 0.2 mg P/mg C/daye Daphnia max conversion efficiency 0.8a half saturation constant of Daphnia ingestion 0.25 mg C/la phosphorus half saturation constant of algae 0.008 mg P/ld Daphnia loss rate 0.25/day

d phosphorus loss rate of algae 0.05/dayDz Effective diffusivity 0.1 m2/dayDu algal particle diffusivity 0.0001 m2/dayDv Daphnia particle diffusivity 0.01 m2/dayDp phosphorus particle diffusivity 0.0001 m2/dayK0 carrying capacity at surface z = 0 0-3 mgC/lκu algal specific light attenuation coefficient 0.0004 m2/mg Cκbg background light attenuation coefficient 0.3− 0.9 m−1

H total depth 20 m

2.2.3 Numerical methods

We conduct series of numerical experiments of Model 2.12 and Model 2.13 in

order to understand the population dynamics under various environmental conditions.

The parameters values are summarized in Table 2.3.

The reaction-advection diffusion equation Model 2.12 and Model 2.13 together

with eq. (2.17) for radiative transport, form a system of nonlinear partial differential

equations which must be solved numerically. Our simulation methods are based

35


on those developed by Dissanayake 2016 [13] for an algae-Daphnia ecosystem with

stoichiometry and diffusion, which were then further extended in Dissanayake at al

2019 [14]. The reader is referred to [14] for a fuller description of the numerical

methods used.

2.2.4 Results

Numerical simulations of Model 2.13 are presented in Figures 2.8 and 2.9 for

varying levels of light K0 and two concentrations of total phosphorus Pt. The simu-

lations represent a one dimensional water column with a depth of 20 meters. Surface

plots for the algae producer population density for varying time and depth are shown

in Figure 2.8. Under low light conditions corresponding to a low surface algae car-

rying capacity K0 = 0.5 mg C/L the population exist at a stable equilibria which

decreases with depth (Figure 2.8(a)). Under higher light conditions corresponding

to a higher surface algae carrying capacity K0 = 1 mg C/L the population exhibits

oscillations near the surface, which dampen to stable equilibria as depth increases

(Figure 2.8(b)).

Figure 2.9 show snapshots of these dynamics for a single time point showing

steady state behaviors for varying depths. The curves represent stable equilibria

and stable limit cycles. The first column of figures considers intermediate values of

Pt = 0.6 mg P, which corresponds to 0.03 mg P/L. The second column of figures

considers high values of Pt = 1.0 mg P, which correspond to 0.05 mg P/L. The

unshaded regions in these figures depict the solutions of the model at equilibria and

the shaded regions depict regions with stable limit cycles.

Population dynamics depend on the level of light at the surface. For a low light

level at the surface corresponding with a low producer carrying capacity K0 = 0.5 mg

36


Time (day)

0 20 40 60 80 100Dep

th (m

)

0.02.55.07.510.012.515.017.520.0 Alga

e po

pula

tion

dens

ity (m

g C/

L)

0.050.100.150.200.250.300.350.40

Pt = 0.6, K0 = 0.5

0.050.100.150.200.250.300.350.40

(a)

Time (day)

0 20 40 60 80 100Dep

th (m

)

0.02.55.07.510.012.515.017.520.0 Alga

e po

pula

tion

dens

ity (m

g C/

L)

0.10.20.30.40.50.60.7

Pt = 0.6, K0 = 1.0

0.10.20.30.40.50.60.7

(b)

Figure 2.8. Numerical simulations of the producer population density for Model (2.13)for K0 = 0.5 mg C/L (a) and K0 = 1.0 mg C/L (b) and intermediate Pt = 0.6 mg P.

C/L the model predicts stable equilibria solutions throughout the entire water column

for both levels of P, see Figures 2.9a and b. The algae and Daphnia population coexist

near the surface. As depth increases, eventually the Daphnia population dies out, near

six meters for both P levels. The algal P:C ratio, Q is lowest near the surface, where

the level of light is highest. Comparing Figure 2.9a with Figure 2.9b, we see that Q

is higher under higher Pt, however the algae and Daphnia population dynamics are

similar. Here, both populations are limited by C, and growth is not limited by P. For

K0 = 0.5 mg C/L Daphnia is limited by food quantity.

For intermediate light levels at the surface corresponding with producer carrying

capacity K0 = 1 mg C/L the model exhibits sustained oscillatory dynamics near

the surface and stable equilibria solutions at low depths for both Pt conditions, see

Figures 2.9c and d. As depth increases, the stable limit cycles collapse between six

and seven meters. Here, the Daphnia are able to persist at deeper depth than the

low light predictions.

37


Under high light levels at the surface corresponding with producer carrying ca-

pacity K0 = 2 mg C/L the model exhibits stable equilibria throughout the entire

water column under intermediate Pt = 0.6 mg P conditions, see Figure 2.9e. The

equilibria observed near the surface consist of high algae densities but constrained

Daphnia densities. These constrained grazer densities are due to stoichiometric con-

straints, as the P:C ratio of algae near the surface is low. Algae near the surface are

exposed to high light levels and therefore make low quality food for the grazer. On

the other hand, stable equilibria observed at lower depths consist of lower algae den-

sities of higher quality, high Q. At these lower depths, despite the lower food quantity

the Daphnia population survives due to high food quality. At very low depths algal

densities become low enough, the Daphnia become limited by food quantity again.

As depth continues to increase, eventually the Daphnia population dies out, near 10

meters.

While the model predicts stable equilibria throughout the entire water column

under high light levels for intermediate P levels, under high light and high Pt = 1.0

mg P conditions, the model exhibits stable limit cycles near the surface, see Figure

2.9f. Near the surface, the Daphnia population obtains very high densities during

the oscillatory dynamics. The limit cycles have large amplitude. Throughout these

oscillations, it is likely that Daphnia experience both C and P limitations, as both food

quality and quantity oscillate. Comparing the region of oscillations (shaded regions

in Figures 2.9c, d, and f.), we can see that under high light levels the oscillations

persist to deeper depth.

38


0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

lation

den

sity (

mgC

/L)

Pt = 0.6, K0 = 0.5

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing ca

pacit

y (m

gC/L)

AlgaeDaphniaQPf

K(z)

(a)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

lation

den

sity (

mgC

/L)

Pt = 1.0, K0 = 0.5

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing ca

pacit

y (m

gC/L)

AlgaeDaphniaQPf

K(z)

(b)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

lation

den

sity (

mgC

/L)

Pt = 0.6, K0 = 1.0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing ca

pacit

y (m

gC/L)

AlgaeDaphniaQPf

K(z)

(c)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

lation

den

sity (

mgC

/L)

Pt = 1.0, K0 = 1.0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing ca

pacit

y (m

gC/L)

AlgaeDaphniaQPf

K(z)

(d)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

lation

den

sity (

mgC

/L)

Pt = 0.6, K0 = 2.0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing ca

pacit

y (m

gC/L)

AlgaeDaphniaQPf

K(z)

(e)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

lation

den

sity (

mgC

/L)

Pt = 1.0, K0 = 2.0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing ca

pacit

y (m

gC/L)

AlgaeDaphniaQPf

K(z)

(f)

Figure 2.9. Numerical simulation snap shots for a fixed time showing steady-statebehavior for Model (2.13) for intermediate Pt = 0.6 mgP (a),(c),(e) and high Pt = 1.0mgP (b),(d),(f). The surface light levels is also varied: K0 = 0.5 mg C/L (a)-(b),K0 = 1.0 mg C/L (c)-(d), and K0 = 2.0 mg C/L (e)-(f). The horizontal axis is depthin meters, so highest light levels occur on the left at the surface. Regions wheresolutions exhibit sustained oscillations are shaded in gray. Unshaded regions depictequilibrium solutions.

2.2.5 Discussion

We developed a stoichiometric producer-grazer model that explicitly tracks the

quantity and the nutritional quality of the producer in time and space. The model is39


formulated by mechanistically accounting for the content of two essential elements C

and P as they vary with depth. The developed model is an extension of the spatial

stoichiometric producer-grazer model by Dissanayake 2016 [13], which assumes algal

is extremely efficient at taking up nutrients. Here, we explicitly track free P in the

environment. Numerical simulations yield rich dynamics where the existence and

stability of equilibria and limit cycles depend on depth (Figures 2.8, 2.9).

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

n de

nsity

(mgC

/L)

Pt = 0.03 mg P/L, K0 = 1.5 mg C/L

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing

capa

city

(mgC

/L)

AlgaeDaphniaQK(z)

(a)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

n de

nsity

(mgC

/L)

Pt = 0.03 mg P/L, K0 = 1.5 mg C/L

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing

capa

city

(mgC

/L)

AlgaeDaphniaQPf

K(z)

(b)

Figure 2.10. Numerical simulations snap shots for a fixed time showing steady-statebehavior for Model (2.12) (a) and Model (2.13) (b) for K0 = 1.5 mg C/L and P = 0.03mg P/L. Regions where solutions exhibit sustained oscillations are shaded in gray.Unshaded regions depict equilibrium solutions.

In order to investigate the impact of tracking free P we compare numerical sim-

ulations of two stoichiometric spatially heterogeneous models, Model 2.12 developed

by Dissanayake 2016 [13] with our modification of light absorption, and our extension

which explicitly tracks environmental P, Model 2.13. Figures 2.10 and 2.11 present a

comparison between Model 2.12 and Model 2.13. The parameter values for light and

P used in these simulations are examples where the two models predict qualitatively

different dynamics. Here, for Model 2.13 we set Pt to correspond with the same level

of total P used in Model 2.12. Under high light K0 = 1.5 mg C/L and intermediate

40


0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

n de

nsity

(mgC

/L)

Pt = 0.035 mg P/L, K0 = 2.5 mg C/L

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing

capa

city

(mgC

/L)

AlgaeDaphniaQK(z)

(a)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.0

0.2

0.4

0.6

0.8

1.0

Popu

latio

n de

nsity

(mgC

/L)

Pt = 0.035 mg P/L, K0 = 2.5 mg C/L

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Carry

ing

capa

city

(mgC

/L)

AlgaeDaphniaQPf

K(z)

(b)

Figure 2.11. Numerical simulations snap shots for a fixed time showing steady-statebehavior for Model (2.12) (a) and Model (2.13) (b) for K0 = 2.5 mg C/L and P =0.035 mg P/L. Regions where solutions exhibit sustained oscillations are shaded ingray. Unshaded regions depict equilibrium solutions.

levels of total P = 0.03 mg P/L Model 2.12 predicts stable equilibria throughout the

entire water column, however Model 2.13 exhibits stable limit cycles near the surface

and stable equilibria at lower depths, see Figure 2.10. Interestingly, under very high

light K0 = 2.5 mg C/L and slightly higher intermediate level of total P = 0.035 mg

P/L the behavior of the predicted population dynamics switches between the two

models, see Figure 2.11. Under these parameter conditions, Model 2.12 now exhibits

sustained oscillatory dynamics near the surface, whereas Model 2.13 has stable equi-

libria throughout the entire water column. Figures 2.10 and 2.11 are two example

regions in parameter space where our Model 2.13 has qualitatively different dynamics

than previous models that neglect to track environmental P loads.

The discrepancies between the predictions of these two models can have impor-

tant implications. Under certain environmental conditions the previous model that

incorporate spatial dynamics but neglect to track aquatic free P predict stable equi-

libria when our model predicts stable limit cycles near the surface. Here the grazer

41


Daphnia population densities can get to low values during the oscillations, where they

are in danger of stochastic extinction (Figure 2.10). During these oscillatory dynamics

the amplitudes of the oscillations become important to avoid possible extinction.

Interestingly, under very high light the previous spatial model that neglects to

track aquatic free P predicts limit cycles near the surface where our model predicts

the existence of stable equilibria (Figure 2.11). Further investigations into these

discrepancies should be conducted. We hypothesize that the dynamics are sensitive

to the resource limitation switch Daphnia experience, where they’re growth is either

nutrient or light limited.

These types of qualitative differences are also seen in the absence of spatial dy-

namics. Wang et al. 2008 [58] demonstrated that explicitly tracking free nutrients

in spatially homogeneous stoichiometric models can yield qualitatively different dy-

namics than spatially homogeneous models that don’t allow for the environmental

nutrient load. Peace et al 2014 [44] also investigated the effects of explicitly tracking

free nutrients in a model that considers the consequences of excess nutrients as well

nutrient limitations. Mechanistically formulated models that track free nutrients, like

Model 2.13 and the models presented in Wang et al. 2008 [58] and Peace et al 2014

[44] can be easily expanded to multiple producers and grazers while maintaining their

structure.

It is important to note that in this manuscript we considered simplified dynamics

for particle transport and focused our efforts on exploring the effects of stoichiometric

constraints on population dynamics over space and time. Future enhancements to

the model should include buoyancy, as plankton can self-regulate buoyancy to seek

nutrients or light.

42


CHAPTER 3

REDUCED-ORDER MODELING

3.1 Introduction

A well-known property of the Singular Value Decomposition (SVD) is that it can

produce optimal low-rank approximations to a matrix [57]. Reduced-Order Modeling

(ROM) is the generalization to this idea that can be use to interpret space-time

simulations in terms of an underlying low-dimensional dynamical system. The general

idea of this technique is that we obtain and record a set of ‘snapshot’ results from the

numerical simulations of a time-dependent PDE; we then use those results to produce

an empirical basis with which we can summarize and visualize full simulation results

in terms of a small set of physically meaningful variables and approximate the full

PDE by a small system of ODEs. The variables in the reduced-order model are the

coefficients in an expansion of the solution as a linear combination of the empirical

basis functions. While we will concentrate on the use of ROM to simplify PDE

simulations, the same idea can be used for many other purposes, for example, to find

the important combinations of variables in a complex reaction network or to model

the most significant fluctuations in a stochastic dynamical system. A brief survey of

the history of this approach to ROM and its connection to methods in statistics can

be found in [48].

3.2 Description of ROM

Consider an initial-boundary-value problem for a vector-valued function ω(x, t, a)

where x is a spatial variable, t is time, and a is a vector of parameters appearing in

the problem. We assume the boundary conditions are homogeneous; if not, a simple

43


change in variables is made to make them homogeneous [48].

For simplicity of presentation, we will restrict ω to be scalar-valued, ignore the

parameter a, and work in one spatial dimension on the interval [0, 1]. Relaxing these

restrictions introduces bookkeeping and notational complexity but does not change

the mathematical issues.

Suppose we have used a discretization method with Nx nodes in space and any

suitable time-stepping method in time to obtain and record ‘snapshots’ of ω at times

tj, j = 1 : Nt, where Nt is the number of timesteps. The snapshots don’t need to

start at the beginning of the simulation; we can, for instance, start recording only

after initial transients have died out. We will also assume the snapshots are evenly

spaced; this assumption is easily relaxed. The nodal values of ω at the jth snapshot

are recorded in a vector wj ∈ RNx ; the value at mesh node i is recorded in element

i of wj, which we’ll call wj,i. We pack the snapshots columnwise into a matrix

W ∈ RNx×Nt ,

W = [w1|w2| · · · |wNt ],

which we will call the snapshot matrix. We note here the slightly confusing, though

ultimately logical, notation: the element Wij is wj,i , not wi,j.

3.2.1 The ROM basis and reconstruction of results

The essential idea will be to find a low dimensional, orthonormal basis through

which we can well approximate (in a sense to be defined) any of the vectors wj. The

simplest way to do this is with the SVD: to find a basis of dimension R, we compute

the SVD W = UΣV ∗, then use the first R left singular vectors ur, r = 1 : R. Error

estimates in Euclidean or Frobenius norms then follow from the properties of the SVD

44


[57].

For a given vector wj, its projection onto the rth basis vector is then cr(tj) =

u∗rwj. The time evolution of the system is described well by the evolution of the

variables cr with time.

However, since our original problem was set in an infinite dimensional Hilbert

space, we need to state optimality conditions and error estimates with norms in that

space rather than in RNx . This will make the calculation of the ROM basis slightly

more complicated than simply taking an SVD.

3.2.1.1 An inner-product and norm appropriate to FEM

Now suppose we have a Finite Element (FE) discretization in space, with Nx

FE basis functions φi(x)Nx

i=1. At time tj , the function ω(x, tj) is approximated by

interpolation in space with the FE basis functions,

ω(x, tj) =Nx∑i=1

wj,iφi(x).

Let u and v be functions represented in this basis, with vectors of nodal values u and

v, respectively. Then the L2 inner product between u and v is

(u, v) =Nx∑m=1

Nx∑n=1

umvn

∫ 1

0

φmφn dx

or, (u, v) = uTSv,

where S ∈ RNx×Nx is the mass matrix: Smn =∫ 1

0φmφn dx. Here we note for later

reference that S is SPD. Hereafter, we will use this inner product and its associated

45


norm,

‖v‖ =√

vTSv.

3.2.1.2 Finding a reduced-order basis

So that we can go on to the use of the reduced-order basis, we present without

proof or derivation the procedure for finding a reduced-order basis. Proof that this

procedure yields an optimal basis will be presented in Section 3.2.3.

Pack the snapshot vectors wj into the columns of a snapshot matrix W . To

construct a reduced-order basis of dimension R, form the matrix B = SWW TS, and

find the R dominant eigenpairs of the generalized eigensystem BΨ = λSΨ. We will

call the set of appropriately normalized eigenvectors ΨrRr=1 the reduced-order basis.

Because B is symmetric (SPD, in fact) and S is SPD, the eigenvectors are orthogonal

in the inner product xTSy; thus the reduced order basis is orthonormal in that inner

product. The eigenvalues λr should be recorded as they will be useful in forming

error estimates.

Notice that when S is the identity, the R dominant eigenpairs of B are the R

leading singular vectors ur, r = 1 : R, and the squares of their associated singular

values σ2r . The naive ROM results from simplifying the inner product. In problems

where finite differences have been used for spatial discretization, there is no natural

inner product and the naive method might as well be used; with a finite element

simulation, an L2 or H1 inner product is appropriate.

The basis vector Ψr ∈ RNx is a vector of nodal values for a continuous function

46


interpolated in the FE basis,

ψr(x) =Nx∑i=1

Ψr,iφi(x).

By construction, the set ψr(x)Rr=1 is orthonormal in the L2 inner product defined

above.

3.2.1.3 Reduced-order reconstruction of existing results

We can use the reduced-order basis to approximate a known solution ω(x, tj) by

a reduced-order approximation

ωR(x, tj) =R∑i=1

cr(tj)ψr(x),

where the coefficients cr(tj) are computed by orthogonal projection,

cr(tj) = (ψr(x), ω(x, tj)) = ΨTr Swj.

Compare this to the naive result, where cr(tj) = u∗rwj; again, the only difference is

the modified inner product.

47


3.2.1.4 Error estimates

The time-integrated L2 error in this reconstruction is

ER2 =

Nt∑j=1

∥∥∥∥∥Nx∑

r=R+1

cr(tj)ψr(x)

∥∥∥∥∥2

=Nt∑j=1

Nx∑r=R+1

cr(tj)2

=Nx∑

r=R+1

Nt∑j=1

(ΨTr Swj

) (wTj SΨr

)=

Nx∑r=R+1

ΨTr SΨr

=Nx∑

r=R+1

λr.

The relative error is then

ERENx

=

√√√√∑Nx

r=R+1 λr∑Nx

r=1 λr.

The fraction of the energy captured by the dimension-R reconstruction is

fR =

∑Rr=1 λr∑Nx

r=1 λr.

Typically, a desired value for fR is chosen (say, 0.99) and then R is selected to meet

that condition. Similar error estimates for low-rank approximations via SVD can be

found in [57].

48


3.2.2 From a PDE to a low-dimensional ODE system

In Section 3.2.1 we saw how to represent existing solutions using a ROM basis.

Now we use the ROM basis to project a PDE onto a low-dimensional approximating

space, yielding a system of ODEs. Suppose our PDE is a reaction-diffusion system

with reaction term σ(u) and diffusivity D,

ut = Duxx + σ(u),

with some homogeneous boundary conditions. As our ROM basis functions ψr(x)

have been formed from computed solutions to this equation, they obey the boundary

conditions by construction.

Approximate the solution by

uR(x, t) =R∑r=1

cr(t)ψr(x),

where the cr(t) are to be determined. Substitute into the equation,

R∑r=1

c′r(t)ψr = DR∑r=1

cr(t)∂2ψr∂x2

+ σ(uR),

multiply by a test function ψq, and integrate

c′q(t) = DR∑r=1

cr(t)

∫ 1

0

ψqψ′′rdx +

∫ 1

0

ψqσ(uR)dx. (3.1)

In the first term on the RHS, integrate by parts and note that boundary terms will

49


vanish because of the homogeneous BCs,

∫ 1

0

ψqψ′′rdx = −

∫ 1

0

ψ′qψ′rdx.

In this expression, expand the ROM basis functions ψ in terms of the FE basis

functions φ, finding

∫ 1

0

ψ′qψ′rdx =

Nx∑m=1

Nx∑n=1

Ψq,mΨr,n

∫ 1

0

φ′mφ′n dx

= ΨTqKΨr,

where K is the usual stiffness matrix for the diffusion operator in the finite element

basis φ.

To put the new system of equations in compact form, we define the vector

c(t) = [c1(t), c2(t), · · · , cR(t)]T .

Create a matrix A ∈ RR×R with elements Aqr = DΨTqKΨr, and define the vector

valued function

b(c(t)) =

∫ 1

0

σ(uR(c))[ψ1(t), ψ2(t), · · · , ψR(t)]Tdx.

Then, equation (3.1) becomes

dc

dt= −Ac + b(c).

50


3.2.3 An optimal basis for least squares approximation

We stated above that a ROM basis of dimension R is obtained by finding the R

dominant eigenvectors of

SWW TSΨ = λSΨ.

In this section we show that this procedure yields a basis that is optimal in the sense

of minimizing the squared error

Nt∑j=1

‖ω(x, tj)− ωR(x, tj)‖2 .

3.2.3.1 Which basis works best for all snapshots?

In the previous subsection we saw how to approximate a single snapshot ω(x, tj)

given an approximating basis ψr . But we don’t want to approximate just one snap-

shot; we want to find good approximations to all snapshots. Therefore, we ask which

orthonormal basis set φrRr=1 will minimize the sum over snapshots of the squared

residuals,

J =Nt∑j=1

‖ω(x, tj)− ωR(x, tj)‖2

=Nt∑j=1

∥∥∥∥∥ω(x, tj)−R∑r=1

cr(tj)ψr(x)

∥∥∥∥∥2

.

Expand out this quantity, using the orthogonality of the ψr’s

J =Nt∑j=1

[(ω(x, tj), ω(x, tj))− 2

R∑r=1

cr(tj)(ω(x, tj), ψr(x)) +R∑r=1

cr(tj)2

],

51


and the values of the c’s obtained via projection,

J =Nt∑j=1

[(ω(x, tj), ω(x, tj))−

R∑r=1

(ω(x, tj), ψr(x))2

],

and then note that

(ω(x, tj), ω(x, tj)) = wTj Swj,

(ω(x, tj), ψr(x)) = ΨTr Swj,

to find that

J =Nt∑j=1

(wTj Swj −

R∑r=1

(ΨTr Swj

)2

).

In each term of the sum, the first term wTj Swj is independent of Ψr, so to minimize

J we must maximize the quantity

J =Nt∑j=1

R∑r=1

(ΨTr Swj

)2

=R∑r=1

Nt∑j=1

(ΨTr Swj

) (wTj SΨr

)=

R∑r=1

ΨTr SWW TSΨr,

subject to the orthonormality constraints

ΨTr SΨq = δrq.

52


We can solve this optimization problem using the method of Lagrange multipliers.

It will turn out that we need only enforce the normalization part of the orthonor-

mality constraints; orthogonality will follow naturally. The Lagrangian including the

normalization constraints is

L =R∑r=1

ΨTr SWW TSΨr −

R∑r=1

λr(ΨTr SΨr − 1

).

Then the adjoint equations are (after cancellation of a common factor of two),

0 =1

2

∂L

∂Ψr

= SWW TSΨr − λrSΨr,

and the state equations are

0 =∂L

∂λr= 1−ΨT

r SΨr.

We see that the optimal basis vectors are the normalized eigenvectors of the general-

ized eigenvalue problem (SWW TS

)Ψr = λrSΨr.

Since both S and SWW TS are SPD, the eigenvalues are real and positive, and the

eigenvectors are orthogonal. Since the goal was to maximize

J =R∑r=1

ΨTr SWW TSΨr =

R∑r=1

λr,

we take the R largest eigenvectors.

53


3.2.4 Beyond the simplest cases

Now we look at the complexities that arise when we break the simplifying as-

sumptions. None of these are show-stoppers, but they complicate the notation and

programming.

3.2.4.1 Complex geometry in multiple spatial dimensions

To adapt to a complex domain Ω ⊂ Rd, replace the elements of S and K by

Smn =

∫Ω

φm(x)ψn(x)dx,

and

Kmn =

∫Ω

∇φm · ∇ψndx.

In the calculation of b(c), do the integrals

bq(c) =

∫Ω

ψq(x)σ(c)dx.

3.2.4.2 Snapshots from runs with multiple parameters

Suppose we run the model with p different choices of parameter vector a :

a1, a2, · · · , ap. For the pth run, record the snapshot matrix Wp. Then form the

aggregate snapshot matrix by columnwise concatenation,

W = [W1 | W2 | · · · | Wp]

and use this matrix in constructing the ROM.

54


3.2.5 Non-homogeneous boundary conditions

As described in [18], for example, a problem with non-homogeneous boundary

conditions can be transformed to a homogeneous problem by writing the unknown

ω(x, t) as ω0(x, t)+ν(x, t) where ω0(x, t) is any sufficiently smooth function that obeys

the BCs and ν(x, t) is a new unknown.

3.2.5.1 BCs independent of time and parameters

In cases where the boundary values are independent of time and model param-

eters, a function ω0(x, t) can be constructed easily from snapshot values. Let wpj be

the jth snapshot vector from the run with parameter set ap. Compute the average

snapshot vector,

ω0 =1

NtP

P∑p=1

Nt∑j=1

wpj ,

and offset all snapshots by subtracting this vector. Then, when reconstructing a

reduced order function, add this vector back in:

ωR(x, t) =Nx∑i=1

w0,iφi(x) +R∑r=1

cr(t)ψr(x).

3.2.5.2 BCs dependent on parameters, independent of time

In this case, offset the columns in the pth parameter set’s snapshot matrix by the

time-averaged snapshot vector

wp0 =

1

Nt

Nt∑j=1

wpj .

Form the aggregate snapshot matrix by concatenation, and generate the ROM basis.

When reconstructing a reduced-order function for a ROM run with parameter set p,

55


add this vector back in:

ωR(x, t) =Nx∑i=1

wp0,iφi(x) +

R∑r=1

cr(t)ψr(x).

If doing a run with parameter values for which no snapshot exists, construct an

appropriate time-average vector by interpolation of the known wp0 vectors.

3.2.5.3 BCs dependent on parameters and time

In this case, every snapshot vector must be offset by a spatial discretization of the

function ω0(x, tj) used to satisfy the BCs. Call this vector wpj . When reconstructing

a reduced-order function for a ROM run with parameter set p, add this vector back

in:

ωR(x, tj) =Nx∑i=1

wpjφi(x) +

R∑r=1

cr(t)ψr(x),

or

ωR(x, t) = ω0(x, t) +R∑r=1

cr(t)ψr(x).

If doing a run with parameter values for which no snapshot exists, or at time t not in

the tabulated time values tj, construct an appropriate offset vector by interpolation

of the known w vectors.

3.2.6 Multiple unknowns

The simplest case is when all unknowns use the same finite element basis set.

Suppose we let the vector-valued unknown ω(x, t) ∈ Rn be

ω(x, t) =[ν1(x, t) ν2(x, t) · · · νn(x, t)

]T.

56


Then the jth snapshot vector is arranged as

wj =[ν1(x1, tj) ν2(x1, tj) · · · νn(x1, tj)| ν1(x2, tj) ν

2(x2, tj) · · · νn(x2, tj)| · · ·]T,

and the mass matrix S is replaced with a block mass matrix with an n × n identity

matrix times every entry in the original mass matrix.

We note that it is also possible to reverse the ordering of the variables, so that all

nodal values for ν1 are listed, then all nodal values for ν2, and so on. The mass matrix

then becomes diag

S, S, · · · , S︸︷︷︸n times

. This is easier to program, but is significantly

less efficient because it leads to more cache misses. Most finite element software will

use by default the variable-fastest rather than node-fastest ordering. With a mixed-

element discretization in which different variables have different finite element basis

functions, the notation becomes significantly more complicated but the basic idea is

the same.

3.3 Mechanistic Model: ROM Approach

Simulations of spatio-temporal behavior of biological systems produce large data

sets that can be difficult to analyze. To further analyze the dynamics of the spatial

stoichiometric mechanistic model developed in Chapter 2 we use reduced-order model-

ing to interpret the simulations in terms of an underlying low-dimensional dynamical

system. The purpose of this approach is to gain biological insight.

3.3.1 ROM basis for mechanistic model

We run simulations of the Mechanistic Model 2.13 for various environmental con-

ditions and build the snapshot matrix W . Figure 3.1(a) shows how fast the singular

57


values of W are dropping off, and Figure 3.1(b) shows that the first five singular

values of W are within 10% of the largest singular value (σ1), and capture the most

dominant behavior of the dynamics of our systems. We consider the first five left

singular vectors as our basis (Figure 3.2). We denote the set of the ROM basis as

φi(z)5i=1.

20 40 60 80 100

10 6

10 5

10 4

10 3

10 2

10 1

100

log(

/1)

Semilog plot of / 1 vs

(a)

2 4 6 8 10 12 14

10 2

10 1

100

log(

/1)

Semilog plot of / 1 vs

(b)

Figure 3.1. Singular values of the snapshot matrix W of Model 2.13. The x-axisrepresents the index ν of the singular value σν , and the y-axis is log (σν/σ1).

3.3.2 Results

To analyze the dynamics we project the current simulation of the Model 2.13 onto

the reduced-order basis. Figure 3.3a shows the projection for low surface irradiance

when K0 = 0.75 and Figure 3.3b shows the projection for intermediate surface irra-

diance when K0 = 1.5. We observe that the projection onto the first basis captures

the most dominant behavior of the dynamics. We also observe that the projection

onto φ4(z) and φ5(z) is negligible compared to the projection onto the first three

bases. Therefore, we conclude that the underlying dynamics of the model are three-

dimensional. We consider a 3D phase plane to analyze the dynamics. Figure 3.4

58


0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.04

0.02

0.00

0.02

0.04Ar

bitra

ry u

nits

1(z)AlgaeDaphniaLightFree P

(a)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.04

0.02

0.00

0.02

0.04

Arbi

trary

uni

ts


(b)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.04

0.02

0.00

0.02

0.04

Arbi

trary

uni

ts


(c)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.04

0.02

0.00

0.02

0.04

Arbi

trary

uni

ts


(d)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Depth (m)

0.04

0.02

0.00

0.02

0.04

Arbi

trary

uni

ts


(e)

Figure 3.2. First five left singular vectors of the snapshot matrix W . The x-axisrepresents the depth and the y-axis represents the population in arbitrary units.

shows the phase plane of the coefficients of the projection onto φ2(z) vs. φ3(z) vs.

φ1(z). For low surface irradiance (Figure 3.4a) we find that the population approaches

a limit point, and for intermediate surface irradiance the population exhibits a limit

cycle (Figure 3.4b).

The bifurcation diagram (Figure 3.5) provides further insight into the effects

of the surface irradiance on population dynamics. We vary the surface irradiance

that corresponds to the surface carrying capacity K0 between 0.25 to 2.5. We take

the maximum and the minimum value of the coefficients of the projections onto the

reduced basis and expect them to be equal for limit points and the opposite of sign

for limit cycles. The bifurcation diagram on the first three bases (Figure 3.5(a-c))

shows qualitatively similar behavior. We find that the dynamics of the model exhibit

a stable limit point for K0 ≤ 0.93, for 0.93 < K0 ≤ 1.73 the dynamics exhibit a stable

59


0 50 100 150 200 250 300 350 400t (days)

10

5

0

5

10

15

Arbi

trary

uni

ts

K0 = 0.75

Projection onto 1(z)Projection onto 2(z)Projection onto 3(z)Projection onto 4(z)Projection onto 5(z)

(a)

0 50 100 150 200 250 300 350 400t (days)

10

5

0

5

10

15

Arbi

trary

uni

ts

K0 = 1.5

Projection onto 1(z)Projection onto 2(z)Projection onto 3(z)Projection onto 4(z)Projection onto 5(z)

(b)

Figure 3.3. Projections of the simulations of model (2.13) onto the reduced basis forK0 = 0.75 (a) and K0 = 1.5 (b). The x-axis represents time t (in days) and the y-axisrepresents the population in arbitrary units.

X

54

32

10

Y0.0

0.20.4

0.60.8

1.01.2

1.4

Z

1

0

1

2

3

4

K0 = 0.75

(a) K0 = 0.75

X

86

42

02

Y1.0

0.50.0

0.51.0

1.52.0

2.5

Z

2024

6

8

K0 = 1.5

(b) K0 = 1.5

Figure 3.4. Phase Plane of the coefficients of the projections onto the reduced basis(φ1(z) − φ3(z)) for K0 = 0.75 (a) and K0 = 1.5 (b). The x-axis represents thecoefficients of the projections onto φ2(z), the y-axis represents the coefficients of theprojections onto φ3(z), and the z-axis represents the coefficients of the projectionsonto φ1(z).

limit cycle, and this limit cycle collapses to a limit point for K0 > 1.73.

3.3.3 Discussion

We used the reduced-order modeling technique to further analyze and interpret

the simulations of the mechanistically derived space-time stoichiometric producer-

60


0.5 1.0 1.5 2.0 2.5K0

6

4

2

0

2

4

6

Coef

ficie

nts o

f the

pro

ject

ion

onto

1(

z) Max. value at last 1 yearMin. value at last 1 year

(a)

0.5 1.0 1.5 2.0 2.5K0

6

4

2

0

2

4

6

Coef

ficie

nts o

f the

pro

ject

ion

onto

2(


(b)

0.5 1.0 1.5 2.0 2.5K0

6

4

2

0

2

4

6

Coef

ficie

nts o

f the

pro

ject

ion

onto

3(


(c)

Figure 3.5. Bifurcation diagram of the model (2.13) for surface irradiance that corre-sponds to surface carrying capacity K0 varying from 0.2 to 2.5. The x-axis representsK0 and the y-axis represents the coefficients of the projection onto φ1(z) (a), φ2(z)(b), and φ3(z) (c).

grazer model (Model 2.13). We built a snapshot matrix from the simulations of the

model for various environmental conditions. The singular value decomposition of the

snapshot matrix reveals that the first five singular values are within 10% of the largest

singular value and capture the most dominant behavior of the dynamics. We consider

the first five left singular vectors as the reduced-order basis, which allows us to do

phase plane and bifurcation analysis for the model partial differential equations. We

find a Hopf bifurcation for the surface light irradiance that corresponds to the surface

61


carrying capacity K0. A similar type of Hopf bifurcation was also observed in the

spatially homogeneous model developed in [58].

62


CHAPTER 4

PRECONDITIONING IMPLICIT RUNGE–KUTTA METHODS FOR

PARABOLIC PDES

4.1 Overview

The reaction-diffusion equations of the spatio-temporal stoichiometric producer-

grazer system eq. (2.13) can be viewed as a stiff system of equations with decay

behavior and have specific stability needs, such as needing A-stable or L-stable time-

stepping methods. Also, generating bifurcation diagrams of the population interaction

Fig. 3.5 using the reduced-order basis require solving the PDE system eq. (2.13)

hundreds of times, which is time consuming. Therefore, higher-order time integrator

would be beneficial in this case.

Certain classes of implicit Runge-Kutta (IRK) methods, such as the Radau IA,

Radau IIA, and Lobatto IIIC methods, provide both L-stability and higher-order

accuracy. However, these methods are not commonly used in PDE discretization

because they lead to large, strongly coupled linear systems that need to be solved at

each time step. An s-stage IRK system has s times as many degrees of freedom as

the systems resulting from backward Euler or implicit trapezoidal rule discretization

applied to the same equation set. For example, if a PDE has been linearized and

then discretized with N degrees of freedom, using an s-stage IRK method leads to an

sN×sN linear system that must be solved at each time step. Iterative methods, such

as GMRES, can be used to solve such linear system; and preconditioning is essential

for their rapid convergence. In this chapter, we will investigate block preconditioning

techniques for the block linear system arising from IRK discretization of parabolic

63


PDEs. This work has recently been published in [47].

4.2 Appropriate Time-Integrator for Parabolic PDEs

In this section, we will discuss stability properties of time-stepping methods and

their importance to PDE discretization. Parabolic PDEs are often approximated by

a large system of stiff ODEs. The concept of a “stiff” system is difficult to define

precisely. A description, if not a rigorous definition, of a stiff system – following those

given by Iserles [29] and by Hairer and Wanner [60]– is

“An ODE system is stiff if its numerical solution by some methods requires a

substantially tiny timestep size to avoid instability”.

While the above concept helps to choose and implement numerical methods, it

does not provide a measure of stiffness by which a given system can be judged. One

such measure for system of ODEs is the stiffness ratio.

Definition 4.2.1. The stiffness ratio of an ODE system y′ = Ay is the ratio of

the magnitudes of the largest and the smallest eigenvalues of A. For a general system

y′ = Φ(t,y) its the ratio of the magnitudes of the largest and the smallest eigenvalues

of the Jacobian matrix ∂Φ∂y

.

In general, we say that if the ODE system has a large stiffness ratio then it is stiff.

However, for a non-linear system, large stiffness ratio may not necessarily indicate

that the system is actually stiff. For more detail on stiff problems and stability of

time-integrators, the reader is referred to [29, 60].

64


4.2.1 Stability of time-integrator

To define various notion of stability of a time-integrator, we consider the following

model initial value problem (IVP):

y′ = λy, Re(λ) < 0 (4.1a)

y(0) = 1. (4.1b)

Definition 4.2.2. The stability function of a time-integrator is the function R(z)

on the complex plane such that, when applied to the model IVP (4.1),

y(k+1) = [R(z)](k+1)y(0), whereλ ∈ C.

Definition 4.2.3. A method is called absolutely stable at htλ if, when applied to

the model IVP (4.1) with a given step-size ht and λ ∈ C, the sequence yn → 0 as

n→∞.

Definition 4.2.4. The Domain of Absolute Stability (DAS) is the set of htλ ∈ C

such that the method is absolutely stable.

Definition 4.2.5. A method is called A-stable if its domain of absolute stability

(DAS) includes the left half of C:

DAS ⊇ C−, where C− := z ∈ C : Re(z) < 0.

Definition 4.2.6. A method is called L-stable if it is A-stable and limz→∞|R(z)| = 0.

For a stiff differential equation with rapid transient, an L-stable method damps

out the transient after a few iterations and converges to the exact solution. On the

65


other hand, an A-stable method forces the numerical solution to honor the transient,

and as a result, oscillates around the exact solution for all timesteps. The following

is an illustration of this effect of a L-stable method presented in [60] and in Josh

Engwer’s thesis [17].

Consider the following stiff IVP, known as the Curtiss-Hirschfelder equation:

u′ + 50u = 50 cos t (4.2a)

u(0) = 0. (4.2b)

The exact solution is

u(t) = −2500

2501e−50t +

2500

2501cos t+

50

2501sin t, (4.3)

where the exponential term represents the rapid transient. Figure 4.1(a) shows that

the solution plots of (4.2) with Trapezoidal method oscillates about the exact solution

whereas Figure 4.1(b) shows Backward Euler method converges after one iteration.

4.2.2 CFL condition

The CFL condition, named after Richard Courant, Kurt Friedrichs, and Hans

Lewy, is a necessary condition for the convergence of a numerical approximation of a

PDE. For a detailed and thorough presentation of CFL condition we refer to [56].

Consider the heat equation

∂u

∂t− ∂2u

∂x2= 0, (4.4)

66


(a) (b)

Figure 4.1. Solution plots of (4.2) with Trapezoidal method (a) and Backward Eulermethod (b). Figure taken from [17].

where u(x, t) is the temperature. The CFL condition for (4.4) is given by

ht ≤1

2h2x.

A-stable methods are not restricted to the CFL condition, we can choose the step

size ht on accuracy considerations only [29]. It is well known that explicit methods

are not A-stable and therefore restricted to the CFL condition.

4.2.3 Second Dahlquist barrier

The following theorem, famously known as the second Dahlquist barrier,

presented in the classic paper by Dahlquist [11] gives an important bound for the

order of convergence of an A-stable linear multistep method.

Theorem 4.2.1. There are no A-stable explicit linear multistep methods. A-stable

implicit multistep methods are 2nd-order at best.

67


Since our goal is to use high-order A-stable or L-stable time-stepping methods for

the solution of parabolic PDEs, we must choose the implicit Runge–Kutta methods

as these methods offer both high-order and A-stability or L-stability.

4.3 Implicit Runge–Kutta Methods

IRK methods are described in detail in, for example, [60]. The general s-stage

IRK method for u′ = f(t, u) requires the solution of s simultaneous equations

Ki = f

(tn + ciht, un + ht

s∑j=1

aijKj

)for i = 1 to s

for the stage variables Ki, i = 1 to s. The stage variables are then used to update

the solution u,

un+1 = un + ht

s∑i=1

biKi.

The coefficients that define a given method are summarized in a Butcher table:

c1 a11 · · · a1s

......

. . ....

cs as1 · · · ass

b1 · · · bs

=

c A

bT .

We list some common IRK methods with their order and stability properties in

Table 4.1. As our goal is to use higher order L-stable methods, throughout this paper

we consider Radau IIA and Lobatto IIIC methods. Of the common L-stable methods,

these provide the highest order for a given number of stages, that is, for a given cost.

68


Table 4.1. Common IRK MethodsIRK Methods Order Stability

Gauss–Legendre (s) 2s A-stableRadau IA (s) 2s− 1 L-stableRadau IIA (s) 2s− 1 L-stableLobatto IIIA (s) 2s− 2 A-stableLobatto IIIC (s) 2s− 2 L-stableMiller DIRK (2) 2 L-stableMiller DIRK (3) 2 L-stableCrouzeix SDIRK(2) 2 L-stableCrouzeix SDIRK(3) 3 L-stable

4.4 Choice of Solvers

In this section, our focus is to discuss methods for solving a linear system

Ax = b, (4.5)

when A is large.

Although direct methods such as LU and Cholesky factorization gives exact so-

lution to (4.5) (neglecting rounding error), these methods are impractical to use for

large system due to the computational cost. Therefore, we will focus on iterative

methods. Krylov subspace methods, such as GMRES (Generalized Minimum Resid-

ual) and CG (Conjugate Gradient), are the most commonly used iterative methods

for (4.5) when A is large and sparse. CG is used when A is symmetric positive definite

(SPD) and GMRES is used when A is not SPD and AT not available [12]. For more

details on the iterative solvers for large-scale problems we refer to the classic book by

Yousef Saad [49].

69


4.5 Preconditioning

Preconditioning transforms a linear system into a mathematically equivalent sys-

tem with improved properties for solution by an iterative method. A good precondi-

tioner can improve both the efficiency and robustness of iterative methods [49], and

sometimes it is necessary for an iterative method to converge at all [12]. The first

step in preconditioning the linear system (4.5) is to find a preconditioning matrix P

that has the following minimum requirements:

1. the eigenvalues of P−1A are clustered together away from zero, and

2. Px = b is easy to solve.

Once a preconditioning matrix P is available there are three ways we can apply the

preconditioner to (4.5). We can apply the preconditioner to the left of (4.5), known

as left preconditioning, leading to the preconditioned system

P−1Ax = P−1b. (4.6)

Alternatively, we can apply the preconditioner to the right of (4.5), known as right

preconditioning:

AP−1y = b, x = P−1y. (4.7)

Finally, a split preconditioning is possible when we have P in a factored form:

P = PLPR; (4.8)

70


the preconditioned system is then given by

P−1L AP−1

R y = P−1L b, x = P−1

R y. (4.9)

This type of split preconditioning is used to preserve the symmetry of the linear

system being solved.

Commonly used preconditioning methods for block systems, such as those that

arise from IRK discretization of parabolic PDEs, are Jacobi preconditioning and the

Gauss-Seidel preconditioning. We discuss this block structure in section 4.6. The Ja-

cobi preconditioner, also known as the Diagonal preconditioner, is simply the diagonal

part of A: PJ = diag(A). Gauss-Seidel preconditioners are triangular precondition-

ers. For the Gauss-Seidel upper triangular preconditioner, the preconditioning matrix

is the upper triangular part of A: PGSU = triu(A). And for the Gauss-Seidel lower

triangular preconditioner, the preconditioning matrix is the lower triangular part of

A: PGSL = tril(A).

These two preconditioners can easily be defined for block systems. For example,

let A be a 2× 2 block system given by

A =

B C

D E

,where B,C,D,E ∈ RN×N . Then the block Jacobi preconditioner is given by the

block diagonal part of A:

PJ =

B 0

0 E

,

71


and the block Gauss-Seidel upper and lower triangular preconditioners are given by

the block upper and lower triangular parts of A respectively:

PGSU =

B C

0 E

,

PGSL =

B 0

D E

.4.6 IRK Methods Applied to Test PDEs

We consider two test problems. The first is a simple heat equation, and the

second is a double-glazing advection-diffusion problem.

4.6.1 Heat equation

As a first simple test problem, we consider the heat equation on some spatial

domain Ω over the time interval [0, T ] with Dirichlet boundary conditions:

ut = ∇2u in Ω× [0, T ]

u = 0 on ∂Ω

u(x, 0) = u0(x).

(4.10)

We first discretize in time using an IRK method. We write un for u(x, nh) and

introduce stage variables Ki = Ki(x) for i = 1 to s. In strong form, the stage

72


equations for an s-stage IRK method are:

Ki = ∇2un + ht

s∑j=1

aij∇2Kj in Ω

Ki = 0 on ∂Ω for i = 1 to s.

We discretize this system of time-independent PDEs using the finite element

method. Adopting test functions Ki ∈ H10 , we obtain a weak form as usual by

multiplying the i-th stage equation by the test function Ki, integrating, and applying

Green’s identity. This leads to the equations:

∫Ω

[KiKi +∇Ki · ∇un + ht

s∑j=1

aij∇Ki · ∇Kj

]dΩ−

∫∂Ω

Kin · ∇un dΓ

−hts∑j=1

aij

∫∂Ω

Kin · ∇Kj dΓ = 0,

which must hold for all Ki ∈ H10 . Note that the boundary terms vanish due to the

boundary restriction of Ki to H10 . Therefore we need to find Ki ∈ H1

0 such that

∫Ω

[KiKi +∇Ki · ∇un + ht

s∑j=1

aij∇Ki · ∇Kj

]dΩ = 0 ∀Ki ∈ H1

0 .

Converting to weak form and discretizing with finite element basis functions φj,

73


we get the following linear system, which must be solved for the stage variables Ki:

M + a11htF a12htF · · · a1shtF

a21htF M + a22htF · · · a2shtF

......

. . ....

as1htF as2htF · · · M + asshtF

K1

K2

...

Ks

= −

Fun

Fun

...

Fun

, (4.11)

where the elements of the matrices M and F are given by

Mkl =

∫φkφl = O

(hdx)

Fkl =

∫∇φk · ∇φl = O

(hd−2x

),

and where d is the spatial dimension of the problem. Thus we have an sN×sN linear

system to solve at each time step; this is the system that we need to precondition.

4.6.2 Double-Glazing advection-diffusion problem

As a second test problem, we consider a model advection-diffusion problem known

as the double-glazing problem. A time-independent version of this problem is de-

scribed in detail in [15]; we modify it here to a time-dependent problem. This is a

simple model for advective-diffusive transport in a cavity Ω = [−1, 1]× [−1, 1], where

74


one wall is hot. The equation and boundary conditions are

ut + (w · ∇)u− ε∇2u = 0 in Ω× [0, T ]

u = 0 on ∂ΩN ∪ ∂ΩW ∪ ∂ΩS

u = 1 on ∂ΩE

u(x, 0) = u0(x),

(4.12)

where ε > 0 and ∂ΩN , ∂ΩW , ∂ΩS and ∂ΩE are the North, West, South, and East

walls of Ω. For initial conditions we take u0(x) = 0 on Ω\∂ΩE; the solution evolves

towards the steady solution described in [15]. The transport is advection dominated

when ε 1 and diffusion dominated when ε & 1. For the wind w(x), we use the

circulating interior flow described in [15]. As before, we discretize in time using an

IRK method then convert the resulting time-independent PDEs to weak form and

discretize in space using stabilized finite elements. This results in a linear system for

the stage variables Ki of the same structural form as (4.11).

4.7 Block Preconditioning IRK Methods for Parabolic PDEs

In this section, we develop a block upper triangular preconditioner and a block

lower triangular preconditioner for systems of the form (4.11).

Let A be the s-stage IRK coefficient matrix, that is, the matrix comprising the

75


elements in the upper-right block of the Butcher table for the given IRK method:

A =

a11 a12 · · · a1s

a21 a22 · · · a2s

......

. . ....

as1 as2 · · · ass

.

Given an m × n matrix B and a matrix C, recall that their Kronecker product

B ⊗ C is given by

B ⊗ C =

b11C b12C · · · b1nC

b21C b22C · · · b2nC

......

. . ....

bm1C bm2C · · · bmnC

.

Using this notation, we can write the block matrix (4.11) arising from the FEM

discretization as

A = Is ⊗M + htA⊗ F,

where Is is the s× s identity matrix.

In this paper, we consider several block preconditioners P of the general form

P = Is ⊗M + htA⊗ F, (4.13)

where A is either a diagonal or triangular matrix referred as the preconditioner coef-

ficient matrix. We can, for example, define a block Jacobi preconditioner PJ and a

76


block Gauss–Seidel preconditioner PGSL using this notation as follows:

PJ = Is ⊗M + htAJ ⊗ F

PGSL = Is ⊗M + htAGSL ⊗ F,(4.14)

where AJ is the diagonal part of A and AGSL is the lower triangular part of A. These

preconditioners and their order optimality have been extensively studied in Mardal

et al. [39] and in G.A. Staff et al. [53]. We now introduce two new LDU -based block

triangular preconditioners for A.

4.7.1 LDU-based block triangular preconditioners

It has been shown in [53] that if all blocks of A are well preconditioned by a

preconditioner P of the form (4.13), then all of the eigenvalues of the preconditioned

system will be clustered, and the condition number of the preconditioned system can

be approximated by κ(P−1A) ≈ κ(A−1A

), where A is the preconditioner coefficient

matrix. In other words, if we make A an effective preconditioner for the Butcher

coefficient matrix A, then P can be expected to be an effective (left) preconditioner

for the system A. (In practice, we use the same motivation in developing left and

right preconditioners.) We begin, therefore, by preconditioning A.

For the development of our preconditioners, we assume that the IRK coefficient

matrix A is invertible and has nonsingular leading principal submatrices. It is proven

in [21] that the Butcher coefficient matrices for Gauss–Legendre, Radau IA, Radau

IIA, and Lobatto IIIC are nonsingular for all s. We have confirmed that the lead-

ing principal submatrices are all nonsingular for the coefficient matrices arising from

Gauss–Legendre, Lobatto IIIC, Radau IA, and Radau IIA IRK methods for all stages

77


that we use in this paper: s = 2 through s = 7. Given these conditions, A can be fac-

tored into A = LDU without pivoting. We compute formally the LDU factorization

A = LDU without pivoting

a11 a12 · · · a1s

a21 a22 · · · a2s

......

. . ....

as1 as2 · · · ass

=

1

l21 1

......

. . .

ls1 ls2 · · · 1

d11

d22

. . .

dss

1 u12 · · · u1s

1 · · · u2s

. . ....

1

and consider ADU = DU and ALD = LD as preconditioners for A. Since A = LDU ,

we have

A = LDU

A(DU)−1 = L

and

(LD)−1A = U.

The eigenvalues of U and L are all one, thus we expect ADU to be a good right

preconditioner and ALD to be a good left preconditioner for A. Since the eigenvalues

(though not the condition numbers) of the product of two matrices do not depend on

the order of the product, we consider both matrices as left and right preconditioners

in our numerical experiments.

Using the condition number motivation from [53], we now define block upper and

78


lower triangular preconditioners for A of the form:

PDU = Is ⊗M + htADU ⊗ F

PLD = Is ⊗M + htALD ⊗ F.(4.15)

4.7.2 Application of the preconditioners

Applying the block Jacobi preconditioner PJ to a vector v is straightforward,

and for an s-stage IRK scheme involves s subsolves. The subsolve for the jth block

has the form (M + htAjjF

)wj = vj.

In practice and for efficiency, we use a single algebraic multigrid (AMG) V-cycle for

each subsolve for the block wj.

Application of the block upper triangular preconditioner PDU is done via back

substitution and again involves s subsolves. The subsolve on the jth block in this

case is of the form

(M + htAjjF

)wj =

[vj −

s∑k=j+1

htAjkFwk

].

In practice, we again use a single AMG V-cycle for each subsolve.

Application of the block lower triangular preconditioners PGSL and PLD is done

via forward substitution and again involves s subsolves. The subsolves on the jth

block in this case are of the form

(M + htAjjF

)wj =

[vj −

j−1∑k=1

htAjkFwk

].

79


As with the previous preconditioners, we use a single AMG V-cycle for each subsolve.

4.7.3 Analysis of the preconditioners

The preconditioners have been chosen because of their intended effect on the

condition number of the preconditioned system. Although, as expected, the condition

number is not always a good indicator of performance for GMRES. In Table 4.2, we

examine the 2-norm condition number for the unpreconditioned system A and for A

preconditioned on the right by PJ , PGSL, PDU , and PLD. The system A is from a

2D heat equation using an s-stage Radau IIA IRK method for s ranging from 2 to

7. In all of these results, we have chosen hx = 2−3 and ht = hp+12s−1x , where p = 2 is

the degree of the Lagrange polynomial basis functions in space. The preconditioners

are constructed exactly. All of the preconditioners significantly reduce the condition

number, with PGSL and PLD giving the most improvement. PGSL gives the greatest

improvement on the condition number for the lower-order stages s = 2 and s = 3,

and PLD gives the greatest improvement for the higher-order stages s = 3 through

s = 7.

Table 4.2. Condition numbers of right-preconditioned matrices with preconditionersP−1J , P−1

GSL, P−1DU , and P−1

LD applied to a 2D heat equation with s-stage Radau IIA

methods. Here, hx = 2−3 and ht = hp+12s−1x , where p = 2 is the degree of the Lagrange

polynomial basis functions in space. Preconditioners are constructed exactly.

s κ(A) κ(AP−1J ) κ(AP−1

GSL) κ(AP−1DU) κ(AP−1

LD)

2 240.37 3.23 1.75 5.32 2.483 502.53 5.66 2.58 11.18 2.664 746.23 8.54 3.63 18.23 3.045 959.16 11.76 5.08 26.53 3.216 1137.24 15.23 7.13 35.97 3.507 1281.47 18.90 10.05 46.48 3.67

80


In Table 4.3, we examine the 2-norm condition number for the unpreconditioned

system A arising from the double-glazing problem, for A preconditioned on the left

by PGSL and PLD, and for A preconditioned on the right by PGSL and PLD. We have

set ε = 0.005, which corresponds to a weakly advection-dominated problem. We are

using an s-stage Radau IIA IRK method for s ranging from 2 to 7. In all of these

results, we have chosen hx = 2−4 and ht = hp+12s−1x , where p = 1 is the degree of the

Lagrange polynomial basis functions in space. The preconditioners are constructed

exactly. As with the heat problem, both of the preconditioners significantly reduce the

condition number. In this table, we omit results for the block Jacobi preconditioner

and our upper triangular preconditioner since, as in the heat test problem, these did

not perform as well as PGSL and PLD. The PLD preconditioner applied on the right

gives greater improvement than PGSL applied on the right for the higher-order stages

s = 4 through s = 7. PLD applied on the left gives the greatest improvement in

condition number for all stages.

Table 4.3. Condition numbers of left-preconditioned and right-preconditioned matri-ces with preconditioners P−1

GSL and P−1LD applied to a 2D double-glazing problem with

ε = 0.005, and with s-stage Radau IIA methods. Here, hx = 2−4 and ht = hp+12s−1x ,

where p = 1 is the degree of the Lagrange polynomial basis functions in space. Pre-conditioners are constructed exactly.

κ(P−1A) κ(AP−1)

s κ(A) κ(P−1GSLA) κ(P−1

LDA) κ(AP−1GSL) κ(AP−1

LD)

2 631.32 1.67 1.26 1.73 2.403 1237.60 2.67 1.53 2.51 2.574 1802.02 4.14 1.80 3.54 2.925 2310.65 6.47 2.05 4.92 3.076 2762.17 10.23 2.25 6.83 3.327 3155.26 16.30 2.44 9.55 3.46

81


With GMRES, the condition number does not always tell the whole story, and it

is also important to consider how well the preconditioner clusters the eigenvalues. In

Figures 4.2 and 4.3, we show eigenvalues plotted in the complex plane for the matrix

A and preconditioned systems. In Figure 4.2 the matrix A is from the 2D heat

problem and this system is right preconditioned with PJ , PGSL, PDU , and PLD. In

Figure 4.3, A is from the 2D double-glazing problem with ε = 0.005 and this system

is left preconditioned with PGSL and PLD. In both figures, we are using an s-stage

Radau IIA IRK method for s ranging from 2 to 5, and we have again chosen hx = 2−3

and ht = hp+12s−1x , where p is the degree of the Lagrange polynomial basis functions in

space; p = 2 for the heat problem and p = 1 for the double-glazing problem. The

preconditioners are constructed exactly. In both cases, the original matrix A has a

number of eigenvalues very near zero. All of the preconditioners succeed in clustering

the eigenvalues farther away from zero. In the heat problem with s = 2, PGSL and

PLD give the tightest clustering and have very similar eigenvalues. But as s increases,

PLD maintains tighter clustering than PGSL. This fact is reflected in our numerical

results presented in the next section. In the double-glazing problem, the original

matrix A also has a number of eigenvalues very near zero. In this problem, both

preconditioners cluster the eigenvalues farther from zero. Our PLD preconditioner

gives the tightest clustering for all stages s.

4.7.4 Comparison between PGSL with optimal coefficients and PLD preconditioners

In G.A. Staff et al. [53], the authors develop an optimal block lower triangular

Gauss–Seidel preconditioner. This block Gauss–Seidel preconditioner is derived such

that the preconditioner coefficients AGSL are computed from the following optimiza-

82


(a) (b)

(c) (d)

Figure 4.2. Eigenvalues of the matrices A, AP−1J , AP−1

GSL, AP−1DU , and AP−1

LD for2D heat problem with Radau IIA s = 2 (a), s = 3 (b), s = 4 (c), and s = 5 (d).The x-axis is the real part and the y-axis is the imaginary part of the eigenvalue.Preconditioners are constructed exactly.

tion problem

minAGSL

κ(A−1GSLA)

s.t. diag (AGSL) = diag (A).

(4.16)

That is, their optimal preconditioner, which we will refer to as PGSL, uses coefficients

that optimize the condition number of the (left) preconditioned system for a block

lower triangular matrix subject to the constraint that the diagonal entries are the

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(a) (b)

(c) (d)

Figure 4.3. Eigenvalues of the matrices A, P−1GSLA, and P−1

LDA for 2D double-glazingproblem with ε = 0.005, and with Radau IIA s = 2 (a), s = 3 (b), s = 4 (c), ands = 5 (d). The x-axis is the real part and the y-axis is the imaginary part of theeigenvalue. Preconditioners are constructed exactly.

same as those in the Butcher matrix A.

In Table 4.4, we examine the 2-norm condition number for A left preconditioned

by PGSL and PLD for 1D and 2D heat problems. For PGSL, we use the optimized

AGSL coefficients given in [53]. We fix hx = 2−3 and ht = hp+1q

x , where p = 2 is the

degree of the Lagrange polynomial basis functions in space and q is the order of the

corresponding IRK method (that is, q = 2s − 1 for Radau IIA and q = 2s − 2 for

Lobatto IIIC). We use Radau IIA with s = 2 through s = 6 and Lobatto IIIC with s =

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2 through s = 4. For the Lobatto IIIC problems, we see that PGSL consistently yields

a lower condition number than PLD for all stages s. For the Radau IIA problems,

however, the PLD preconditioner yields a lower condition number than the optimized

preconditioner PGSL for all stages s. In [53], the authors comment that they may not

obtain the global minimum to (4.16) with the optimization process they use. This may

indeed be true in some of their results, but is not the only issue. Their optimization

is subject to the constraint diag (AGSL) = diag (A). Thus they are not optimizing

over all lower triangular preconditioners, but rather over an affine subspace of the set

of all possible lower triangular preconditioners. The values used in PLD, which have

different diagonal elements, give a lower condition number in the Radau IIA case.

Table 4.4. Condition numbers of the left-preconditioned system with preconditionersPGSL (PGSL with optimal coefficients) and PLD for various IRK methods applied

to 1D and 2D heat problems. Here, hx = 2−3 and ht = hp+1q

x , where p = 2 is thedegree of the Lagrange polynomial in space and q is the order of the correspondingIRK method (that is, q = 2s − 1 for Radau IIA and q = 2s − 2 for Lobatto IIIC).Preconditioners are constructed exactly.

1D 2D

κ(P−1GSLA) κ(P−1

LDA) κ(P−1GSLA) κ(P−1

LDA)

R IIA 2 1.58 1.26 1.60 1.26R IIA 3 1.94 1.50 1.93 1.52R IIA 4 2.18 1.74 2.17 1.77R IIA 5 2.37 1.95 2.40 1.98R IIA 6 3.01 2.14 3.18 2.18

L IIIC 2 1.48 2.68 1.54 2.78L IIIC 3 3.20 6.68 3.32 6.94L IIIC 4 4.92 10.43 5.24 10.93

Again we note that for GMRES, a lower condition number does not always

indicate a superior preconditioner. In Figures 4.4 and 4.5 we see that PLD more

85


effectively clusters the eigenvalues for both Radau IIA and Lobatto IIIC for stages

s = 2, 3, 4 and 5.

(a) (b)

(c) (d)

Figure 4.4. Eigenvalues of the matrix A, AP−1GSL, and AP−1

LD for 2D heat problemwith Radau IIA s = 2 (a), s = 3 (b), s = 4 (c), and s = 5 (d). The x-axis is thereal part and the y-axis is the imaginary part of the eigenvalue. Preconditioners areconstructed exactly.

4.8 Numerical Experiments

To test the performance of our preconditioners we consider the two model prob-

lems described in Section 4.6: a 2D heat equation problem and the 2D advection-

diffusion double-glazing problem. For both we use s-stage Radau IIA or Lobatto

86


(a) (b)

Figure 4.5. Eigenvalues of the matrix A, AP−1GSL, and AP−1

LD for 2D heat problemwith Lobatto IIIC stages s = 3 (a) and s = 4 (b). The x-axis is the real part andthe y-axis is the imaginary part of the eigenvalue. Preconditioners are constructedexactly.

IIIC methods for time discretization. For the heat equation we discretize in space

using Galerkin finite elements with piecewise quadratic basis functions on triangular

meshes. For the double-glazing problem we use streamwise upwind Petrov-Galerkin

discretization [15] with piecewise linear basis functions, again on triangular meshes.

Dirichlet boundary conditions are imposed using Nitsche’s method [54, 31].

Production of the finite element matrices is carried out with the Sundance finite

element toolkit [37]. The method of manufactured solutions is used to produce an

exact solution to each linear subproblem so that we can check the error in addition

to the residual.

A common practice in analysis of methods for spatiotemporal problems is to

choose ht ∼ hx and increase the polynomial order of spatial discretization in lockstep

with the order of time discretization. Here, we use a different approach. We use

low-order elements (p = 1 or p = 2) throughout, and then for each hx we choose a

87


timestep ht such that the spatial discretization error and temporal global truncation

error are comparable. With a Radau s-stage method, we set hp+1x = h2s−1

t so that the

timestep is ht = hp+12s−1x , and with a Lobatto IIIC method, we set hp+1

x = h2s−2t so that

the timestep is ht = hp+12s−2x . By holding the spatial order p fixed, we are simulating

conditions in which a modeler uses a high order integrator to obtain results of a

specified accuracy with larger timestep and smaller computational cost.

We solve the linear system (4.11) using preconditioned GMRES. We construct

our preconditioners as described in Section 4.7 using one AMG V-cycle for each

subsolve. We use AMG from the IFISS software package [52]. When applied to the

advection-diffusion equation, multilevel algorithms and smoothers must be handled

with care [15]. In these experiments we use the strategy of ensuring that the SUPG

discretization is stabilized on the coarse mesh, and use the smoothed aggregation

smoother from IFISS. All results are computed in MATLAB on a machine with an

Intel Core i7 1.80 GHz (Turbo Boost up to 4.00 GHz) processor and 8.00 GB RAM.

In Section 4.7, we examined condition numbers and eigenvalue distributions for

the preconditioners PJ , PGSL, PGSL, PDU , and PLD applied to the 2D heat equation

(4.10) and the double-glazing advection-diffusion problem (4.12). In this section, we

investigate the convergence performance of our preconditioner compared to the others.

We start with the 2D heat equation (4.10) in Section 4.8.1 and compare iteration

counts and timing results for GMRES preconditioned on the right by PJ , PGSL,

PDU , and PLD for various problem sizes and varying orders of IRK methods. We

then examine the robustness of the PGSL and PLD preconditioners for a fixed spatial

resolution as we vary the time step size. We also compare our preconditioner with the

optimized preconditioner PGSL. We conclude the heat equation results by examining

88


the relative errors achieved with each of the preconditioners. In Section 4.8.2 we give

iteration count and timing results for the best performing of the preconditioners, PGSL

and PLD, applied to the double-glazing advection-diffusion problem (4.12). As with

the heat equation results, we first compare methods for varying spatial resolutions

and various order IRK methods. We then fix the spatial resolution and compare the

robustness the methods as we vary the time step size.

4.8.1 Heat Equation Results

We first test the performance of the preconditioners PJ , PGSL, PDU , and PLD,

on our 2D heat problem (4.10).

4.8.1.1 Comparison results between PJ , PGSL, PDU , and PLD preconditioners

In Table 4.5 we report iteration counts and timing results for right-preconditioned

GMRES to converge with relative tolerance 1.0× 10−8 for varying mesh sizes hx and

varying number of stages (varying order). Throughout, all of the triangular precondi-

tioners outperform the block Jacobi preconditioner PJ , as expected. The LDU -based

block upper triangular preconditioner PDU and the block Gauss–Seidel preconditioner

PGSL perform similarly, with PGSL performing slightly better, especially for the higher

stage methods. However, overall, the LDU -based block lower triangular precondi-

tioner PLD performs better than all of the other preconditioners in both iteration

count and timing. For 2-stage Radau IIA, PLD and PGSL have similar performance,

with PLD performing slightly better on the smaller problems and PGSL performing

slightly better on the larger problems. But as the number of stages increases (that

is, with increasing order), the performance of PLD improves over the other precondi-

tioners giving consistently lower iteration counts and timing. For a given number of

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stages, all of the triangular preconditioners scale well with problem size, exhibiting

little to no growth in iteration count with increasing problem size. However the PLD

preconditioner has the least increase in cost as the number of stages increases.

We have also tested all four preconditioners for Lobatto IIIC methods. Table

4.6 shows iteration counts and timing results for right-preconditioned GMRES to

converge with a relative tolerance of 1.0× 10−8 for the 2D heat problem (4.10) using

Lobatto IIIC methods in time and quadratic finite elements in space. For the Lobatto

IIIC methods, the PLD preconditioner outperforms all of the other preconditioners

both in iteration count and timing, with more significant improvements for larger

problems and larger number of stages (higher order).

Because a common time-stepping solution method for (4.10) is an adaptive strat-

egy where the time step increases as the steady-state is approached, we also investigate

the effectiveness of our preconditioner for a fixed spatial resolution while varying the

time step size. In Table 4.7, we report iteration counts and timing results for left

and right-preconditioned GMRES to converge with a relative residual tolerance of

1.0 × 10−8 for the 2D heat problem with 2-stage and 7-stage Radau IIA methods.

In this table, we have fixed the spatial resolution at h−1x = 128 and we vary the

time step size ht between 0.05 and 5.0. Both the PGSL preconditioner and our PLD

preconditioner are quite robust with respect to varying time step size. Both precon-

ditioners perform very similarly (and robustly) for the lower-order s = 2 method;

for the higher-order s = 7 method, both preconditioners are robust with respect to

the varying time step size, but our PLD preconditioner outperforms PGSL as a left

preconditioner and as a right preconditioner for all values of the time step.

90


Table 4.5. Iteration counts and elapsed time (times in seconds are shown in paren-theses) for right-preconditioned GMRES to converge with relative residual tolerance1.0 × 10−8 for a 2D heat problem with s-stage Radau IIA methods with precondi-

tioners PJ , PGSL, PDU , and PLD. Here we choose ht = hp+12s−1x , where p = 2 is the

degree of the Lagrange polynomial in space. Preconditioners are approximated usingone AMG V-cycle for each subsolve.

stage h−1x DOF PJ PGSL PDU PLD

s=2

8 450 14 (0.03) 8 (0.01) 7 (0.01) 7 (0.01)16 1922 14 (0.05) 8 (0.02) 7 (0.02) 7 (0.02)32 7938 14 (0.10) 8 (0.06) 7 (0.06) 7 (0.05)64 32,258 14 (0.40) 7 (0.19) 7 (0.19) 7 (0.19)128 130,050 14 (1.48) 7 (0.80) 7 (0.83) 7 (0.82)

s=3

8 675 23 (0.06) 10 (0.05) 10 (0.03) 9 (0.03)16 2883 22 (0.06) 10 (0.03) 10 (0.03) 8 (0.02)32 11,907 21 (0.17) 10 (0.09) 10 (0.08) 8 (0.07)64 48,387 22 (0.67) 9 (0.29) 10 (0.32) 8 (0.26)128 195,075 21 (2.90) 8 (1.12) 9 (1.25) 8 (1.12)

s=4

8 900 32 (0.10) 13 (0.08) 13 (0.07) 10 (0.04)16 3844 31 (0.11) 12 (0.05) 13 (0.05) 10 (0.04)32 15,876 30 (0.31) 12 (0.13) 12 (0.13) 10 (0.12)64 64,516 29 (1.16) 11 (0.47) 12 (0.52) 9 (0.39)128 260,100 27 (5.24) 11 (2.10) 12 (2.29) 9 (1.73)

s=5

8 1125 42 (0.10) 15 (0.08) 16 (0.08) 11 (0.06)16 4805 40 (0.23) 15 (0.08) 16 (0.08) 11 (0.06)32 19,845 39 (0.50) 13 (0.18) 15 (0.20) 11 (0.17)64 80,645 35 (1.80) 13 (0.70) 15 (0.81) 11 (0.61)128 325,125 33 (8.44) 12 (2.98) 14 (3.49) 11 (2.75)

s=6

8 1350 53 (0.15) 18 (0.08) 19 (0.08) 12 (0.04)16 5766 50 (0.31) 17 (0.11) 18 (0.11) 12 (0.08)32 23,814 46 (0.73) 16 (0.27) 18 (0.29) 12 (0.19)64 96,774 42 (2.66) 15 (1.02) 17 (1.14) 12 (0.81)128 390,150 38 (12.13) 14 (4.27) 17 (5.41) 12 (3.56)

s=7

8 1575 62 (0.22) 21 (0.10) 23 (0.10) 13 (0.09)16 6727 57 (0.40) 20 (0.15) 22 (0.16) 13 (0.09)32 27,783 52 (0.98) 19 (0.37) 21 (0.40) 13 (0.26)64 112,903 48 (3.64) 18 (1.49) 20 (1.61) 12 (0.98)128 455,175 44 (17.06) 17 (6.29) 20 (7.24) 12 (4.42)

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Table 4.6. Iteration counts and elapsed time (times in seconds are shown in paren-theses) for right-preconditioned GMRES to converge with relative residual tolerance1.0 × 10−8 for a 2D heat problem with s-stage Lobatto IIIC methods with precon-

ditioners PJ , PGSL, PDU , and PLD. Here we choose ht = hp+12s−2x , where p = 2 is the


stage h−1x DOF PJ PGSL PDU PLD

s=2

8 450 17 (0.01) 10 (0.01) 7 (0.03) 7 (0.01)16 1922 18 (0.03) 10 (0.02) 7 (0.02) 8 (0.02)32 7938 19 (0.13) 10 (0.07) 8 (0.06) 8 (0.06)64 32,258 19 (0.47) 10 (0.26) 7 (0.19) 8 (0.22)128 130,050 18 (2.19) 10 (1.21) 7 (0.86) 8 (1.00)

s=3

8 675 27 (0.05) 12 (0.03) 11 (0.02) 10 (0.02)16 2883 29 (0.07) 12 (0.03) 11 (0.03) 10 (0.03)32 11,907 27 (0.22) 12 (0.10) 11 (0.10) 10 (0.08)64 48,387 29 (0.96) 11 (0.36) 11 (0.34) 10 (0.35)128 195,075 27 (3.89) 11 (1.54) 10 (1.41) 9 (1.27)

s=4

8 900 40 (0.07) 16 (0.05) 15 (0.05) 12 (0.05)16 3844 40 (0.14) 15 (0.06) 15 (0.06) 12 (0.05)32 15,876 39 (0.41) 14 (0.15) 14 (0.15) 12 (0.13)64 64,516 38 (1.58) 14 (0.60) 13 (0.57) 11 (0.47)128 260,100 37 (7.58) 13 (2.46) 13 (2.44) 11 (2.06)

s=5

8 1125 55 (0.08) 19 (0.03) 19 (0.03) 13 (0.02)16 4805 55 (0.26) 18 (0.09) 18 (0.08) 13 (0.07)32 19,845 51 (0.57) 17 (0.20) 17 (0.22) 13 (0.16)64 80,645 49 (2.61) 16 (0.85) 16 (0.84) 12 (0.64)128 325,125 45 (12.01) 15 (3.64) 15 (3.69) 12 (2.96)

4.8.1.2 Comparison results between PGSL with optimal coefficients and PLD

preconditioners

In this section we compare our LDU -based block lower triangular preconditioner

PLD with the optimal block lower triangular Gauss–Seidel preconditioner introduced

in [53]. PGSL was developed using coefficients optimized to reduce the condition

92


Table 4.7. Iteration counts and elapsed time (times in seconds are shown in parenthe-ses) for left-preconditioned and right-preconditioned GMRES to converge with pre-conditioned relative residual tolerance 1.0× 10−8 for a 2D heat problem with s-stageRadau IIA methods with preconditioners PGSL and PLD. Here we keep h−1

x = 128fixed and vary ht from 0.05 to 5.0. Preconditioners are approximated using one AMGV-cycle for each subsolve.

left prec. GMRES right prec. GMRES

stage ht PGSL PLD PGSL PLD

s = 2

0.05 8 (0.85) 7 (0.71) 7 (0.63) 7 (0.63)0.1 8 (0.78) 7 (0.73) 6 (0.55) 7 (0.63)0.5 7 (0.74) 7 (0.71) 6 (0.56) 7 (0.63)1.0 7 (0.70) 7 (0.70) 6 (0.56) 7 (0.64)5.0 7 (0.72) 8 (0.78) 7 (0.63) 7 (0.65)

s = 7

0.05 24 (7.38) 14 (4.24) 19 (5.43) 12 (3.49)0.1 24 (7.76) 14 (4.53) 18 (5.62) 12 (3.70)0.5 23 (7.42) 14 (4.49) 17 (5.34) 13 (4.05)1.0 23 (6.77) 14 (4.56) 17 (4.89) 13 (4.00)5.0 21 (6.74) 15 (4.93) 18 (5.64) 13 (4.17)

number of the left-preconditioned system P−1GSLA.

In Table 4.8, we present results similar to those in Table 4.5. The problem set

up is the same, but here we use left preconditioning for all of the preconditioners

since the coefficients of PGSL were optimized for left preconditioning, and we include

an additional column for PGSL. The first thing we note is that although the idea of

optimizing coefficients to reduce the condition number of the preconditioned system

is a perfectly sensible idea, for this problem the unoptimized PGSL performs better

than the optimized PGSL except on the smallest problems. For all numbers of stages,

PGSL performs better than PGSL for problems sizes h−1x = 32, 64, and 128. We

observe in this table that our PLD has the best performance overall. Although it

performs slightly worse as a left preconditioner than it did in Table 4.5 as a right

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Table 4.8. Iteration counts and elapsed time (times in seconds are shown in parenthe-ses) for left-preconditioned GMRES to converge with preconditioned relative residualtolerance 1.0×10−8 for a 2D heat problem with s-stage Radau IIA methods with pre-

conditioners PJ , PGSL, PGSL, PDU , and PLD. Here we choose ht = hp+12s−1x , where p = 2

is the degree of the Lagrange polynomial in space. Preconditioners are approximatedusing one AMG V-cycle for each subsolve.

stage h−1x DOF PJ PGSL PGSL PDU PLD

s=2

8 450 14 (0.02) 9 (0.02) 9 (0.01) 7 (0.01) 7 (0.01)16 1922 16 (0.03) 9 (0.02) 10 (0.02) 8 (0.02) 7 (0.02)32 7938 16 (0.10) 9 (0.06) 10 (0.06) 8 (0.05) 7 (0.05)64 32,258 17 (0.38) 9 (0.24) 10 (0.27) 8 (0.23) 7 (0.21)128 130,050 18 (2.01) 9 (1.16) 10 (1.24) 8 (1.03) 7 (0.94)

s=3

8 675 23 (0.06) 11 (0.04) 13 (0.03) 10 (0.04) 9 (0.03)16 2883 25 (0.05) 12 (0.03) 13 (0.03) 11 (0.03) 9 (0.02)32 11,907 25 (0.20) 12 (0.09) 14 (0.10) 11 (0.08) 9 (0.07)64 48,387 27 (0.77) 12 (0.36) 14 (0.43) 11 (0.34) 9 (0.31)128 195,075 28 (3.65) 12 (1.65) 14 (1.91) 11 (1.56) 9 (1.32)

s=4

8 900 32 (0.07) 15 (0.06) 18 (0.05) 14 (0.06) 11 (0.04)16 3844 34 (0.13) 15 (0.06) 19 (0.07) 14 (0.06) 11 (0.05)32 15,876 36 (0.38) 15 (0.17) 19 (0.21) 14 (0.16) 11 (0.13)64 64,516 36 (1.52) 15 (0.70) 19 (0.84) 14 (0.64) 11 (0.51)128 260,100 36 (6.51) 15 (2.88) 19 (3.63) 14 (2.74) 11 (2.20)

s=5

8 1125 44 (0.11) 18 (0.08) 22 (0.06) 17 (0.07) 12 (0.06)16 4805 45 (0.21) 18 (0.08) 22 (0.10) 17 (0.08) 12 (0.06)32 19,845 46 (0.61) 18 (0.26) 22 (0.28) 17 (0.23) 12 (0.17)64 80,645 47 (2.21) 18 (0.90) 22 (1.18) 17 (0.93) 12 (0.74)128 325,125 47 (11.15) 18 (4.41) 22 (5.45) 17 (4.26) 12 (3.00)

s=6

8 1350 55 (0.14) 22 (0.08) 24 (0.07) 20 (0.07) 13 (0.04)16 5766 55 (0.32) 22 (0.14) 24 (0.16) 21 (0.14) 14 (0.09)32 23,814 56 (0.93) 22 (0.38) 24 (0.40) 20 (0.34) 14 (0.24)64 96,774 56 (3.77) 21 (1.49) 24 (1.69) 20 (1.42) 13 (0.94)128 390,150 56 (15.89) 21 (6.21) 23 (6.89) 20 (5.63) 13 (3.98)

preconditioner, it nonetheless achieves the lowest iteration count and lowest timing

of all of the preconditioners for all numbers of stages and all problem sizes.

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In Table 4.9, we present results similar to those in Table 4.8 but using Lobatto

IIIC methods. The problem set up is again the same, using left preconditioning for all

of the preconditioners. The results are very similar to those in the previous table. The

unoptimized PGSL performs better than the optimized PGSL on all of these problems.

Our PLD preconditioner has the best performance overall.

Table 4.9. Iteration counts and elapsed time (times in seconds are shown in parenthe-ses) for left-preconditioned GMRES to converge with preconditioned relative residualtolerance 1.0× 10−8 for a 2D heat problem with s-stage Lobatto IIIC methods with

preconditioners PJ , PGSL, PGSL, PDU , and PLD. Here we choose ht = hp+12s−2x , where

p = 2 is the degree of the Lagrange polynomial in space. Preconditioners are approx-imated using one AMG V-cycle for each subsolve.

stage h−1x DOF PJ PGSL PGSL PDU PLD

s=2

8 450 18 (0.02) 11 (0.02) 18 (0.02) 7 (0.02) 8 (0.02)16 1922 20 (0.04) 12 (0.02) 20 (0.03) 8 (0.02) 8 (0.02)32 7938 21 (0.15) 12 (0.09) 21 (0.14) 8 (0.06) 8 (0.06)64 32,258 21 (0.56) 12 (0.34) 21 (0.56) 8 (0.24) 8 (0.23)128 130,050 21 (2.40) 12 (1.49) 21 (2.50) 8 (1.05) 8 (1.05)

s=3

8 675 27 (0.06) 14 (0.04) 25 (0.04) 11 (0.03) 11 (0.03)16 2883 32 (0.08) 14 (0.04) 27 (0.06) 12 (0.03) 11 (0.03)32 11,907 33 (0.23) 15 (0.11) 28 (0.20) 12 (0.09) 11 (0.09)64 48,387 35 (1.06) 15 (0.49) 30 (0.98) 12 (0.40) 11 (0.37)128 195,075 37 (3.89) 15 (1.87) 30 (3.76) 12 (1.67) 11 (1.58)

s=4

8 900 41 (0.09) 18 (0.06) 31 (0.07) 15 (0.06) 13 (0.05)16 3844 44 (0.13) 18 (0.06) 35 (0.11) 16 (0.06) 13 (0.05)32 15,876 46 (0.51) 18 (0.20) 35 (0.38) 16 (0.18) 13 (0.15)64 64,516 49 (2.14) 18 (0.80) 37 (1.66) 16 (0.74) 13 (0.60)128 260,100 49 (8.93) 18 (3.23) 36 (6.60) 16 (3.11) 13 (2.53)

It should be noted that since the results in Tables 4.8 and 4.9 are using left

preconditioning, the relative residuals are actually preconditioned relative residuals.

So the norms of the various preconditioners could be having an effect on the iteration

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counts. Since all of our numerical tests were run using the method of manufactured

solutions, in Tables 4.10 and 4.11 we examine the relative errors for each method.

In Table 4.10 we report the relative errors for various numbers of stages for Radau

IIA and Lobatto IIIC with the five preconditioners PJ , PGSL, PGSL, PDU , and PLD

all used as left preconditioners. The problem size is fixed at h−1x = 128, which is the

largest of the problems we considered, and ht = hp+1q

x , where p = 2 is the degree of the

Lagrange polynomial in space and q is the order of the corresponding IRK method

(that is, q = 2s − 1 for Radau IIA and q = 2s − 2 for Lobatto IIIC). The stopping

criteria is the same as in all previous tables. With left preconditioning, problems with

Radau IIA and Lobatto IIIC methods and all numbers of stages yielded very similar

relative errors.

Table 4.10. Relative error for left-preconditioned GMRES converging to a precondi-tioned relative residual tolerance of 1.0 × 10−8 for a 2D heat problem with variousIRK methods with preconditioners PJ , PGSL, PGSL, PDU , and PLD. Here, hx = 2−7

and ht = hp+1q

x , where p = 2 is the degree of the Lagrange polynomial in space andq is the order of the corresponding IRK method (that is, q = 2s − 1 for Radau IIAand q = 2s− 2 for Lobatto IIIC). Preconditioners are approximated using one AMGV-cycle for each subsolve.

IRK method PJ PGSL PGSL PDU PLDR IIA 2 7.1× 10−9 6.1× 10−9 1.4× 10−9 1.8× 10−9 4.0× 10−9

R IIA 3 8.5× 10−9 6.9× 10−9 7.9× 10−9 9.4× 10−9 6.4× 10−9

R IIA 4 4.6× 10−8 8.5× 10−9 9.1× 10−9 3.8× 10−8 2.9× 10−9

R IIA 5 5.1× 10−8 9.0× 10−9 8.1× 10−9 4.6× 10−8 5.4× 10−9

R IIA 6 7.7× 10−8 1.0× 10−8 8.3× 10−9 9.8× 10−8 7.9× 10−9

L IIIC 2 8.0× 10−9 1.8× 10−9 8.0× 10−9 3.9× 10−9 6.6× 10−9

L IIIC 3 2.0× 10−8 6.7× 10−9 1.0× 10−8 1.8× 10−8 1.1× 10−8

L IIIC 4 5.7× 10−8 1.2× 10−8 7.9× 10−9 2.3× 10−8 8.9× 10−9

In Table 4.11 we report the relative errors for various numbers of stages for Radau

IIA and Lobatto IIIC with the four preconditioners PJ , PGSL, PDU , and PLD this

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time all used as right preconditioners. We do not include the optimized GSL-based

preconditioner PGSL since it is optimized for left preconditioning. The problem size

is again fixed at h−1x = 128, and as before ht = h

p+1q

x , where p = 2 is the degree of the

Lagrange polynomial in space and q is the order of the corresponding IRK method

(that is, q = 2s − 1 for Radau IIA and q = 2s − 2 for Lobatto IIIC). The stopping

criteria is the same as in all previous tables. With right reconditioning, our PLD

preconditioner achieves almost an order of magnitude better relative error on almost

all of the runs.

Table 4.11. Relative error for right-preconditioned GMRES converging to a relativeresidual tolerance of 1.0 × 10−8 for a 2D heat problem with various IRK methods

with preconditioners PJ , PGSL, PDU , and PLD. Here, hx = 2−7 and ht = hp+1q

x , wherep = 2 is the degree of the Lagrange polynomial in space and q is the order of thecorresponding IRK method (that is, q = 2s − 1 for Radau IIA and q = 2s − 2 forLobatto IIIC). Preconditioners are approximated using one AMG V-cycle for eachsubsolve.

IRK method PJ PGSL PDU PLDR IIA 2 1.5× 10−6 1.4× 10−6 2.8× 10−7 3.1× 10−8

R IIA 3 1.5× 10−5 1.5× 10−5 9.0× 10−6 2.8× 10−7

R IIA 4 2.7× 10−5 4.6× 10−6 8.3× 10−6 6.9× 10−7

R IIA 5 3.8× 10−5 1.1× 10−5 1.5× 10−5 1.3× 10−7

R IIA 6 5.6× 10−5 2.0× 10−5 9.5× 10−6 8.2× 10−8

R IIA 7 1.7× 10−6 3.9× 10−7 8.3× 10−7 2.4× 10−7

L IIIC 2 2.6× 10−7 1.3× 10−7 2.5× 10−7 2.1× 10−8

L IIIC 3 1.2× 10−5 8.1× 10−6 9.2× 10−6 6.9× 10−6

L IIIC 4 2.5× 10−5 1.3× 10−5 1.6× 10−5 1.4× 10−6

L IIIC 5 6.1× 10−5 1.3× 10−5 4.0× 10−5 1.2× 10−6

4.8.2 Advection-Diffusion Equation Results

In this section, we examine the performance of the preconditioners PGSL and

PLD, on the 2D double-glazing advection-diffusion problem (4.12). For the double-

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glazing problem, we only present results for the PGSL and PLD preconditioners since

in all of our previous experiments they performed best. We apply both as left precon-

ditioners in these results, but our results are similar when they are applied as right

preconditioners.

In Table 4.12, we report iteration counts and elapsed time for left-preconditioned

GMRES to converge with preconditioned relative residual tolerance 1.0×10−8 for the

double-glazing problem (4.12) with s-stage Radau IIA methods using the precondi-

tioners PGSL and PLD. We vary the number of stages from s = 2 to s = 7 for problems

with spatial resolution h−1x = 64 and 128. Here we choose ht = h

p+12s−1x , where p = 1

is the degree of the Lagrange polynomial in space. We present results for ε = 0.04,

which is diffusion dominated and ε = 0.005, which is weakly advection dominated.

Preconditioners are approximated using one AMG V-cycle for each subsolve.

For the lowest-order problems with s = 2, PGSL and our PLD preconditioner

perform similarly. For the diffusion dominated problem, ε = 0.04, PLD outperforms

PGSL for all of the higher-order problems, s = 3 through s = 7, with the improved

performance being more pronounced on the highest-order problems. For the weakly

advection-dominated problem, ε = 0.005, the results are more mixed. For the smaller

problems of each order, h−1x = 64, PLD takes fewer iterations in most cases, but takes

longer in compute time in a few cases. For the largest higher-order problems, s = 4

through s = 7 with h−1x = 128, PLD performs better than PGSL in both iteration

count and timing.

As we did for the heat equation, in Table 4.13, we investigate the robustness of the

preconditioners for a fixed spatial resolution while varying the time step size. In this

table, we present iteration counts and timing results for left-preconditioned GMRES

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Table 4.12. Iteration counts and elapsed time (times in seconds are shown in parenthe-ses) for left-preconditioned GMRES to converge with preconditioned relative residualtolerance 1.0× 10−8 for the double-glazing problem with s-stage Radau IIA methods

with preconditioners PGSL and PLD. Here we choose ht = hp+12s−1x , where p = 1 is the


ε = 0.04 ε = 0.005

stage h−1x ht DOF PGSL PLD PGSL PLD

s = 264 0.0625 33282 9 (0.50) 7 (0.43) 11 (0.89) 11 (0.83)128 0.0394 132098 9 (1.72) 7 (1.44) 10 (3.74) 10 (3.84)

s = 364 0.1895 49923 14 (1.13) 10 (0.84) 17 (1.92) 16 (1.83)128 0.1436 198147 14 (3.96) 9 (2.73) 15 (8.23) 14 (9.71)

s = 464 0.3048 66564 18 (1.95) 12 (1.28) 22 (3.22) 21 (3.04)128 0.25 264196 18 (6.47) 11 (4.44) 20 (16.83) 17 (15.17)

s = 564 0.3969 83205 22 (2.62) 13 (1.58) 26 (4.54) 25 (4.29)128 0.3402 330245 22 (10.38) 13 (6.45) 25 (22.75) 21 (22.33)

s = 664 0.4695 99846 25 (3.94) 14 (2.33) 31 (6.34) 28 (6.51)128 0.4139 396294 27 (15.84) 14 (10.59) 30 (34.10) 23 (33.38)

s = 764 0.5274 116487 30 (5.39) 16 (2.81) 36 (8.53) 30 (9.04)128 0.4740 462343 31 (20.89) 15 (10.34) 36 (55.84) 25 (36.33)

to converge with a preconditioned relative residual tolerance of 1.0× 10−8 for the 2D

double-glazing problem with s-stage Radau IIA methods using the preconditioners

PGSL and PLD. We present results for a lower-order method s = 2 and a higher-order

method s = 7 and for ε = 0.04, which is diffusion dominated, and ε = 0.005, which

is weakly advection dominated. In this table, we keep h−1x = 128 fixed and vary the

time step size ht from 0.05 to 5.0. Preconditioners are approximated using one AMG

V-cycle for each subsolve.

For the diffusion-dominated problem with the lower-order method, our PLD pre-

99


conditioner mildly outperforms PGSL for all time step sizes, and both preconditioners

are robust with respect to varying the time step size. For the higher-order method,

both preconditioners take more iterations and more time as the step size increases,

but PLD grows less dramatically than PGSL and takes fewer iterations and less time

for all time step sizes.

For the weakly advection-dominated problem, ε = 0.005, PLD performs very

similarly to PGSL on the lower-order problem, s = 2, and both preconditioners are

robust with respect to the varying time step size. For the higher-order method, both

preconditioners take more iterations and more time as the step size increases, but

PLD grows much less dramatically than PGSL and takes fewer iterations and less time

for all time step sizes.

Table 4.13. Iteration counts and elapsed time (times in seconds are shown in parenthe-ses) for left-preconditioned GMRES to converge with preconditioned relative residualtolerance 1.0× 10−8 for a 2D double-glazing problem with s-stage Radau IIA meth-ods with preconditioners PGSL and PLD. Here we keep h−1

x = 128 fixed and vary htfrom 0.05 to 5.0. Preconditioners are approximated using one AMG V-cycle for eachsubsolve.

ε = 0.04 ε = 0.005

stage ht PGSL PLD PGSL PLD

s = 2

0.05 9 (1.85) 7 (1.50) 11 (3.95) 10 (4.12)0.1 10 (1.96) 7 (1.43) 11 (4.24) 11 (5.01)0.5 11 (2.05) 8 (1.76) 12 (4.64) 11 (4.64)1.0 10 (2.08) 7 (1.47) 13 (4.76) 11 (4.72)5.0 10 (1.95) 8 (1.69) 12 (5.63) 12 (5.43)

s = 7

0.05 26 (18.39) 13 (9.78) 27 (41.80) 18 (28.56)0.1 27 (22.09) 14 (9.80) 28 (43.18) 21 (29.72)0.5 32 (24.97) 15 (12.29) 36 (53.60) 25 (40.76)1.0 37 (30.81) 17 (13.88) 45 (76.24) 26 (36.33)5.0 46 (35.33) 20 (16.14) 68 (109.26) 28 (36.66)

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4.9 Discussion

In this chapter, we have introduced a new preconditioner for the large, struc-

tured systems appearing in implicit Runge–Kutta time integration of parabolic PDE

problems. Our preconditioner is based on a block LDU factorization, and for scala-

bility, we have used a single AMG V-cycle on all subsolves. We have compared our

preconditioner PLD in condition number and eigenvalue distribution to other pre-

conditioners, and have demonstrated its effectiveness on two test problems, the heat

equation and the double-glazing advection-diffusion problem. We have found that it

is scalable (hx-independent) and yields better timing results than other precondition-

ers currently in the literature: block Jacobi, block Gauss–Seidel, and the optimized

block Gauss–Seidel method of [53]. It is also robust with respect to varying time step

sizes for a fixed spatial resolution. We ran experiments with implicit Runge–Kutta

stages up to s = 7, and have found that the new preconditioner outperforms the

others, with the improvement becoming more pronounced as spatial discretization is

refined and as temporal order is increased.

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CHAPTER 5

CONCLUSION

In this dissertation, we developed population models that include both spatial

heterogeneity and stoichiometry, explored a reduced-order modeling technique to an-

alyze and interpret the huge data sets produced from the simulations of the model,

and proposed preconditioners for high performance computation of the model with

implicit Runge-Kutta methods.

We discussed space-time population models incorporating ecological stoichiome-

try in chapter 2. In section 2.1, we developed a spatially heterogeneous stoichiomet-

ric producer-grazer (algae-Daphnia) model with two grazer genotypes to understand

genotypic selection. We conducted numerical experiments with the model to inves-

tigate selection processes and co-persistence of grazer genotypes under various envi-

ronmental conditions such as varying light level and nutrient availability. Numerical

simulations of our model yield rich dynamics where genotypic selection depends on

depth. We also observed cases where both Daphnia genotypes co-persist at the same

depth at limit cycles and at equilibria. This work has been published in [14].

In section 2.2, we developed a space-time stoichiometric algae-Daphnia model

that explicitly tracks the nutrient content (C and P). We again conducted several

numerical experiments varying light level and nutrient availability. We observed

qualitatively different dynamics than the space-time stoichiometric model that ne-

glects to track nutrient content explicitly. In particular, under certain environmental

conditions, our model predicts stable limit cycles whereas the model that neglects to

explicitly track nutrient content shows stable equilibria. This work has been published

102


in [46].

In chapter 3, we explored reduced-order modeling to analyze and interpret simu-

lations of the mechanistically derived space-time population model derived in section

2.2. We began this chapter with a description of the Reduced-Order modeling tech-

nique then applied it to the mechanistic model (2.13). We find that the most dominant

behavior of the model dynamics can be captured by five basis vectors. Reduced-order

modeling also allows us to investigate bifurcations of the PDE model for surface light

irradiance.

We began chapter 4 with a discussion on the choice of time-integrator for parabolic

PDEs, and choice of solver and preconditioning technique for the linear system arising

from discretization. We then developed block preconditioners for the block system

arising from implicit Runge-Kutta discretization of parabolic PDEs. Our block pre-

conditioners are based on an LDU−factorization of the IRK coefficient matrix. We

tested the effectiveness of our preconditioners on two test problem: the heat equation

and the advection-diffusion double-glazing problem. Numerical experiments showed

that our LDU -based block lower triangular preconditioner performs better than other

preconditioners currently in the literature. This work has been published in [47].

103


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