Stability of the diagonal waves for discretizations of the advection equation in multidimensions

Stability of the diagonalwaves for discretizations ofthe advection equation in

multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves

Multidimensional spatialstencils

Stability of the multistepschemes

Stability of Runge-Kuttaschemes

Numerical test

Stability of the diagonal waves for discretizations ofthe advection equation in multidimensions

Adrian SescuAbdollah A. Afjeh Ray Hixon Carmen Sescu

University of Toledo, MIME Department, Toledo, OH

SEARCDE, 2008


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Outline of Topics

Short introduction

Parasitic waves

Multidimensional spatial stencils

Stability of the multistep schemes

Stability of Runge-Kutta schemes

Numerical test


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Short introduction

Computational Aeroacoustics (CAA), Large Eddy Simulation (LES) andDirect Numerical Simulation (DNS) are relying on high-order differenceschemes because they are less time consuming, and can be easily adaptedto boundary conditions.

There is a need to build more accurate, high-resolution schemes able tocapture waves over a large range of wavelegths.

Some (most) of the high-resolution schemes perform well inone-dimension, but have issues in multidimensions due to numericalanisotropy.

Waves propagating on the diagonal direction (with respect to the gridlines) are less stable.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Parasitic waves

We start with the initial-value problem in Rd × [0,∞):

∂u

∂t= c∇u; (1)

u(r, 0) = u0(r) (2)

where d is the number of dimensions, r = (x1, ..., xd ) is the vector ofspatial coordinates, c = (cx1 ... cxd ) is the velocity vector,∇ = (∂/∂x1 ... ∂/∂xd )T , u(x1, ..., xd , t) is a scalar function and u0(r)is a periodic function.Let Ω = (x1, ..., xd ), 0 < x1 < l1, ..., 0 < xd < ld be a domain in the realspace Rd with l1 ... ld chosen such that there exist a real non-negativenumber h = l1/N1 = ... = ld/Nd called space step; N1, ..., Nd arenon-negative integers. Next, introduce a uniform equally-spaced grid(x1)i1,...,id = (i1h, ..., idh), ..., (xd )i1,...,id = (i1h, ..., idh), 0 ≤ i1 ≤ N1, ...,0 ≤ id ≤ Nd , and a gridfunction vi1,...,id .


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Parasitic waves

Let’s consider the one dimensional case (d = 1) first:

∂u

∂t=

∂u

∂x1; (3)

where cx1 = 1, and the initial condition

u(x1, 0) = u0(x1) (4)

A centered spatial discretization with arbitrary order of accuracy will leadto

dvi

dt=

1

h

ν=MX

ν=−M

aνEνi · vi (5)

where

Eνi · vi = vi+ν (6)


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Parasitic waves

Using the Fourier-Laplace transform with frequency ω and wavenumber ξ,we get the dispersion relations:

exact numerical

ω = ξ ω =2

h

ν=MX

ν=1

aν sin(νhξ) (7)

The exact and numerical group velocity defined as cg = dω/dξ are

cg = 1 cg = 2ν=MX

ν=1

νaν cos(νhξ) (8)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

ξh

num

eric

al w

aven

umbe

r, (ξ

h)*

exact

2nd order

4th order

6th order

8th order

0 0.5 1 1.5 2 2.5 3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

ξh

num

eric

al g

roup

vel

ocity

, (ξh

)*

exact

2nd order

4th order

6th order

8th order


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Parasitic waves

0 0.5 1 1.5 2 2.5 3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

ξh

num

eric

al g

roup

vel

ocity

, (ξh

)*

exact

2nd order

4th order

6th order

8th order

−100 −80 −60 −40 −20 0 20 40 60 80 100−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u(x)

exact4th order

−100 −80 −60 −40 −20 0 20 40 60 80 100−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u(x)

exact6th order


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Parasitic waves

In two dimensions (d = 2) the advection equation is:

∂u

∂t= cos θ

∂u

∂x1+ sin θ

∂u

∂x2; (9)

where cx1 = cos θ and cx2 = sin θ, and the initial condition

u(x1, x2, 0) = u0(x1, x2) (10)

A centered spatial discretization with arbitrary order of accuracy will leadto

dvi

dt=

1

h

2

4

ν=MX

ν=−M

aν

“

cos θEνi · vi,j + sin θEν

j · vi,j

”

3

5 (11)

where

Eνi · vi,j = vi+ν,j ; Eν

j · vi,j = vi,j+ν ; (12)


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Parasitic waves

The numerical dispersion relation for the semidiscretization is

ω =2

h

ν=MX

ν=1

aν [cos θ sin(νhξ) + sin θ sin(νhη)] (13)

where η is the wavenumber component on the y-direction; it is anon-linear function of wavenumber. The exact dispersion relation isω = ξ cos θ + η sin θ which is a linear function of wavenumbers. Thenumerical group velocity is a vector and is defined ascg = (dω/dξ dω/dη)T

cg = 2

cos θν=MX

ν=1

νaν cos(νhξ) sin θν=MX

ν=1

νaν cos(νhη)

!T

(14)

as opposed to the exact group velocity cg = (cos θ sin θ)T . The exact

group velocity magnitude is |cg | =p

cos2 θ + sin2 θ = 1, whereas thenumerical group velocity magnitude

|cg | = 2

v

u

u

u

tcos2 θ

"

ν=MX

ν=1

νaν cos(νhξ)

#2

+ sin2 θ

"

ν=MX

ν=1

νaν cos(νhη)

#2

(15)is dependent on the direction of propagation θ.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Multidimensional spatial stencils

Consider a Cartesian grid with uniform, equally-spaced pointsxi,j = (ih, jh) and yi,j = (ih, jh) in two dimensions, withi , j = 0,±1,±2, .... The space step is denoted by h and is the same for alldirections. Let v(xi,j , yi,j ) be a gridfunction in two, also denoted by vi,j .The discretizations of the x and y derivatives in two dimensions will bewritten as

„

∂v

∂x

«

i,j

=1

h(1 + β)

ν=MX

ν=−M

aν

„

Eνi +

β

2Dx

«

· vi,j (16)

„

∂v

∂y

«

i,j

=1

h(1 + β)

ν=MX

ν=−M

aν

„

Eνj +

β

2Dy

«

· vi,j (17)

where the operators Dx · and Dy · are defined as

Dx · =“

Eνi Eν

j + E−νi Eν

j

”

·; Dy · =“

Eνi Eν

j + Eνi E−ν

j

”

· (18)

and, as before

Eνi · vi,j = vi+ν,j ; Eν

j · vi,j = vi,j+ν . (19)

β is called the isotropy corrector factor.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


Consider the semidiscrete approximation of the advection equation:

dU

dt= F (U), 0 ≤ t < ∞ (20)

where, in the general case, F is a discrete operator acting on the vector U.In the present case F (U) = AU, where A is a matrix of dimension[N × N]; N is equal to the number of grid points, except those wherephysical boundary conditions are imposed. Only one equationcorresponding to a single grid point is considered,

du(t)

dt= F (u(t)), 0 ≤ t < ∞ (21)

where the spatial indices (i1, ..., id ) were dropped for simplicity. Thegeneral form of a linear multistep method applied to equation (9) is givenby

KX

ν=0

ανun−ν+1 = k

"

KX

ν=0

βνF (un−ν+1)

#

(22)

where k is the time step and n is the time index. They are explicit ifβ0 = 0 and implicit otherwise. We’ll consider the Leap Frog scheme inmultidimensions and the Adams-Bashford method.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


The Leap-Frog scheme in two dimensions using multidimensional stencilsis:

vn+1i,j = vn−1

i,j +2σ

1 + β

ν=MX

ν=−M

aν

»

Eνi + Eν

j +β

2(Dx + Dy )

–

· vni,j (23)

where σ = ck/h, and cx1 = cx2 = c (diagonal waves). If we apply theFourier inversion formula given by

vi,j =1√2π

Z π/h

−π/h

Z π/h

−π/he Ih(iξ+jη)v(ξ, η)dξdη (24)

where I =√−1, ξ ∈ [−π/h, π/h] and η ∈ [−π/h, π/h] are the Cartesian

components of the wavenumber and v(−π) = v(π), we get the equationfor the Fourier amplitudes:

vn+1(ξ, η) = vn−1(ξ, η) + 2σvn(ξ, η)ν=MX

ν=−M

aνHν(hξ, hη) (25)

where

Hν(hξ, hη) = e Iνhξ+e Iνhη+1

2β[2e Iνh(ξ+η)+e Iνh(ξ−η)+e Iνh(−ξ+η)] (26)


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


The quantity

g(hξ, hη, k, h) =vn+1(ξ, η)

vn(ξ, η)(27)

is called the amplification factor and is the amount that shows how theamplitude of each frequency in the solution vn(ξ, η) is amplified as themarching in time proceeds. After simplifications:

ekω = e−kω + 2σν=MX

ν=−M

aνHν(hξ, hη) (28)

and the amplification factor g(hξ, hη, k, h) = ekω is given by

ekω = −IσF (hξ, hη) ±q

1 − σ2F (hξ, hη)2 (29)

where

F (hξ, hη) = 2ν=MX

ν=1

aν Im[Hν(hξ, hη)] (30)

and Im stands for imaginary part.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


Theorem (Von Neumann). A difference scheme (with constantcoefficients) is stable in a stability region Λ if and only if there is aconstant K (independent of hξ, hη and σ) such that

|g(hξ, hη, k, h)| ≤ 1 + Kk (31)

with (k, h) ∈ Λ.If g(hξ, hη, k, h) is independent of k and h, the stability condition can bereplaced with the restricted stability condition

|g(hξ, hη)| ≤ 1 (32)

The stability restrictions for diagonal waves, corresponding to bothclassical and multidimensional stencils are:

Table: Stability restrictions for teh diagonal waves (σ = ck/h, |cx | = |cy |.

Stencil Conv. stencils MultiD stencils MultiD stencils(for isotropy) (max. alowable)

ST3 σ ≤ 0.70711 σ ≤ 0.84933 σ ≤ 1.41421ST5 σ ≤ 0.51530 σ ≤ 0.57601 σ ≤ 1.03060ST7 σ ≤ 0.44585 σ ≤ 0.47746 σ ≤ 0.89170ST9 σ ≤ 0.40859 σ ≤ 0.42680 σ ≤ 0.81718


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


Adams-Bashford method

The general form of a Adams-Bashford method applied to advectionequation is given by

un+1 = un + k

"

KX

ν=0

βνF (un−ν)

#

(33)

The stability regions can be determined using the boundary locus method.The polynomials α(z) and β(z)are given by

α(z) = 1 − z; β(z) =K

X

ν=1

βνzν (34)

where z is given by

z = 2Iν=MX

ν=1

aν(σx sin νhξ + σy sin νhη) (35)

for conventional spatial stencils (I =√−1).


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


The stability regions are determined by mapping the unit circleΘ ∈ [0, 2π] : w = exp(IΘ) into the complex z-plane

w 7−→ 1 − w

β1w + β2w2 + ... + βKwK(36)

Table: Stability limits for some Adams-Bashford schemes.

AB method Stability limitAB3 0.723AB4 0.430AB5 0.224AB6 0.114

−0.4 −0.2 0 0.2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8(a)

real(z)

imag

(z)

−0.3 −0.2 −0.1 0−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5(b)

real(z)

imag

(z)

−0.15 −0.1 −0.05 0−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25(c)

real(z)

imag

(z)

−0.1 −0.05 0

−0.1

−0.05

0

0.05

0.1

0.15(d)

real(z)

imag

(z)

Figure: Stability regions: a) AB3; b) AB4; c) AB5 and d) AB6.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


The stability restriction for Adams-Bashford methods in combination withconventional, centered stencils is given by

2ν=MX

ν=1

aν(σx sin νhξ + σy sin νhη) ≤ (AB)lim (37)

where (AB)lim is the stability limit fo the AB scheme. The stabilityrestriction can also be written as

σx + σy ≤ (AB)lim

(ξmaxh)∗(38)

For Adams-Bashford scheme in combination with multidimensional stencilsthe stability restriction becomes

(1 + β)σx + σy ≤ (1 + β)(AB)lim

(ξmaxh)∗(39)

if cx ≥ cy , and

σx + (1 + β)σy ≤ (1 + β)(AB)lim

(ξmaxh)∗(40)

if cy ≥ cx .


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


On the diagonal direction (cx = cy = c), the stability restrictions in termsof the time step:

k ≤ (AB)lim

(ξmaxh)∗h

2|c| (41)

and

k ≤ (AB)lim

(ξmaxh)∗(1 + β)h

(2 + β)|c| (42)

−20

2−2

02

0

1

2

hξ

(a2)

hη

sqrt

[(hξ

)*2+

(hη)

*2]

−20

2−2

02

0

1

2

hξ

(a2)

hη

sqrt

[(hξ

)*2+

(hη)

*2]

Figure: Numerical dispersion relation surfaces: conventional and multidimensionalstencils.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


The classical Runge-Kutta algorithm is given by:

F1 = F (un), (43)

Fν = F (un + cνkFν−1)), ν = 2, ..., s (44)

un+1 = un + ks

X

ν=1

bνFν (45)

where s is the number of stages, k is the time step and the coefficients cν

and bν are given below for fourth order scheme.

Table: Coefficients of classical fourth order Runge-Kutta method with ν = 1, .., 4.

RK4cν bν

0 1/61/2 1/31/2 1/3

1 1/6


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


A similar condition (as for the Adams-Bashford scheme) is hold

2ν=MX

ν=1

aν(σx sin νhξ + σy sin νhη) ≤ (RK)lim (46)

where (RK)lim is the stability limit coresponding to the specificRunge-Kutta method (see table below).

Table: Stability limits for some Runge-Kutta methods.

ERK method Stability limitLDDRK6 of Hu et al. 1.75

RK4 classic 2.83LDD56 of Stanescu and Habashi 2.85

LDD46 of Calvo et al. 3.82RKo6s of Bogey and Bailly 3.95RKM of Mead and Renaut 4.89

HALE67 of Hixon et al. 5.53


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test


Again:

σx + σy ≤ (RK)lim

(ξmaxh)∗(47)

or in terms of the time step

k ≤ CFLh

|cx | + |cy |(48)

By using multidimensional stencils, the stability restriction becomes:

(1 + β)σx + σy ≤ CFL(1 + β) (49)

or in terms of the time step,

k ≤ CFL(1 + β)h

(1 + β)|cx | + |cy |(50)

For diagonal waves (cx = cy = c), as for Adam-Bashford schemes, thedifference is given by the term (2β + 2)/(β + 2). It’s easy to verify thatlimβ→∞[(2β + 2)/(β + 2)] = 2 showing that the time step can be twiceas large.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Numerical test

Consider the initial-value problem in R2 × [0,∞),

∂u

∂t= cx

∂u

∂x+ cy

∂u

∂y(51)

with the initial condition,

u(x , y , 0) = e−ln2

„

(x−x0)2+(y−y0)2

σ2

«

(52)

where cx and cy are the components of the advection velocity. We choosecx = cy = 1, x0 = y0 = 0 and σ = 0.06, so that the problem can be solvedin the domain [−0.5, 0.5] × [−0.5, 0.5] with periodic boundary conditions:

u(0.5, y , t) = u(−0.5, y , t) (53)

u(x , 0.5, t) = u(x ,−0.5, t) (54)

The domain is discretized using Nx × Ny equally-spaced points. The timemarching is performed using Leap-Frog scheme and variousAdam-Bashford and Runge-Kutta methods.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Numerical test

−0.50

0.5

−0.5

0

0.50

0.5

1

x=y

(a)

z

u(x,

x,z)

−0.50

0.5

−0.5

0

0.50

0.5

1

x=y

(b)

z

u(x,

x,z)

−0.50

0.5

−0.5

0

0.50

0.5

1

x=y

(c)

z

u(x,

x,z)

−0.50

0.5

−0.5

0

0.50

0.5

1

x=y

(d)

z

u(x,

x,z)

Figure: Solution u(x, x, z) at t = 1 using multidimensional stencils for the secondthree dimensional problem: a) exact, b) ST3, c) ST5, d) ST7.


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Numerical test

Table: Ratio between the CPU time using the conventional and the CPU time usingmultidimensional selected stencils - two values are included: one for isotropy and theother corresponding to maximum allowable time step.

a)Leap Frog time marching scheme b)RK4 time marching scheme

Stencil Isotropy Maximum Stencil Isotropy MaximumST3 1.21 1.94 ST3 1.18 1.89ST5 1.02 1.75 ST5 1.01 1.71ST7 0.83 1.47 ST7 0.85 1.45ST9 0.71 1.22 ST9 0.73 1.20

c)LDD46 time marching scheme d)LDDRK6 time marching scheme



multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Numerical test

Table: Ratio between the CPU time using the conventional and the CPU time usingmultidimensional selected stencils - stability restrictions for isotropy and maximumallowable time step, respectively.

a)AB3 time marching scheme b)AB4 time marching scheme


c)AB5 time marching scheme

Stencil Isotropy Max. allowableST3 1.17 1.93ST5 1.01 1.73ST7 0.80 1.41ST9 0.73 1.18


multidimensions

Adrian Sescu

Outline

Short introduction

Parasitic waves




Numerical test

Thank you!

Thank you!

Stability of the diagonal waves for discretizations of the advection equation in multidimensions

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Transcript of Stability of the diagonal waves for discretizations of the advection equation in multidimensions