i NUMERICAL SMOOTHNESS OF ENO AND WENO ...

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NUMERICAL SMOOTHNESS OF ENO AND WENO SCHEMES FOR NONLINEARCONSERVATION LAWS

Jian Wu

A Thesis

Submitted to the Graduate College of Bowling GreenState University in partial fulfillment ofthe requirements of for the degree of

MASTER OF ARTS

August 2011

Committee:

Tong Sun, Advisor

So-Hsiang Chou

Steven Seubert

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c⃝2011

Jian Wu

All Rights Reserved

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AbstractDr. Tong Sun, AdvisorIn this study we create smoothness indicators which quantitively measure the smoothness ofnumerical approximations to solutions of nonlinear conservation laws. We perform numericaltests for these smoothness indicators applied to numerical approximations obtained by apply-ing ENO and WENO schemes. Based on our numerical results, we believe that first thesesmoothness indicators can measure the numerical smoothness of numerical solutions, and sec-ond these smoothness indicators can be used to compare the smoothness of various ENO andWENO schemes. Thus, we believe that these indicators will be useful for the error analysis ofENO and WENO schemes for nonlinear conservation laws.

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AcknowledgmentI would like to thank Dr. Tong Sun to encourage me to write a thesis for my master’s degree,and thank him for directing me to finally finish this thesis.

Contents

1 Introduction 1

2 ENO and WENO 42.1 Reconstruction and Approximation in 1D . . . . . . . . . . . . . . . . . . . . . 42.2 ENO and WENO Approximations in 1D . . . . . . . . . . . . . . . . . . . . . . 52.3 ENO and WENO Schemes for 1D Conservation Laws . . . . . . . . . . . . . . . 13

3 Numerical Smoothness 163.1 Error propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Numerical smoothmess and error analysis . . . . . . . . . . . . . . . . . . . . . 173.3 Smoothness indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Conclusion 71

v

List of Figures

3.1 Third order reconstruction, using Lax-Friedichs flux . . . . . . . . . . . . . . . 203.2 Third order reconstruction, using Godunov flux . . . . . . . . . . . . . . . . . . 223.3 D1 at t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 D1 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 D1 at t = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 D1 at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.7 D1 at t = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.8 zoomed D1 near the shock at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . 293.9 D2 at t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.10 D2 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.11 D2 at t = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.12 D2 at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.13 D2 at t = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.14 Zoomed D2 near the shock at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . 363.15 J1 at t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.16 J1 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.17 J1 at t = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.18 J1 at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.19 J1 at t = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.20 log∆x(J1 +∆x6) at t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.21 log∆x(J1 +∆x6) at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.22 log∆x(J1 +∆x6) at t = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.23 log∆x(J1 +∆x6) at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.24 log∆x(J1 +∆x6) at t = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.25 J2 at t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.26 J2 at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.27 J2 at t = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.28 J2 at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.29 J2 at t = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.30 log∆x(J2 +∆x6) at t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.31 log∆x(J2 +∆x6) at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.32 log∆x(J2 +∆x6) at t = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.33 log∆x(J2 +∆x6) at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.34 log∆x(J2 +∆x6) at t = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

vi

LIST OF FIGURES vii

3.35 Cell shifts r at t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.36 Cell shifts r at t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.37 Cell shifts r at t = 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.38 Cell shifts r at t = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.39 Cell shifts r at t = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.40 WENO reconstruction, using Lax-Friedrichs flux . . . . . . . . . . . . . . . . . 673.41 D1, t=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.42 D2, t=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.43 D3, t=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.44 D4, t=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.45 D5, t=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.46 D6, t=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Chapter 1

Introduction

This thesis is to study the numerical solutions of the nonlinear conservation laws

ut + f(u)x = 0. (1.0.1)

We are primarily concerned with the well-known ENO and WENO schemes, focusing on theirconvergence and error analysis.

Before the early 80s, the mainstream of numerical methods solving conservation laws isapplying linear schemes, such as the first order upwind scheme, the Lax-Wendroff scheme,the Lax-Friedrichs scheme, and so on. However, all of the linear schemes suffer from one ofthe two major drawbacks. First, the numerical diffusion of some schemes produces smearedfronts instead of shocks. Second, all the second or higher order schemes suffer from “GibbsPhenomenon”, and the oscillations brought by “Gibbs Phenomenon” do not decay in magni-tude when the mesh is refined. Due to these two major drawbacks, nonlinear schemes weredeveloped starting from the early 80s, such as TVD (Total Variation Diminishing) scheme(Harten in 1983 [1]), ENO (Essentially Non-Oscillatory) scheme (Harten, Engquist, Osher andChakravarthy in 1987 [11]) and WENO (Weighted Essentially Non-Oscillatory) scheme (C.-W.Shu in 1990 [21]).

Finite difference schemes are related with finite volume schemes. They are based on inter-polations of discrete data using polynomials or other simple functions. In the approximationtheory, it is well known that the wider the stencil, the higher the order of accuracy of theinterpolation, if the function being interpolated is smooth inside the stencil. Traditionally,one applies fixed stencil interpolations to finite difference methods. For example, to obtainan interpolation for cell i to third order accuracy, the infomation of the three cells i − 1, iand i + 1 can be used, to build a second order interpolation polynomial. In other words, onealways chooses one cell to the left, one cell to the right, plus the center cell itself, regardless ofwhere the cell is in the domain. This works well for globally smooth problems. The resultingschemes is linear for linear PDEs, hence stability can be easily analyzed by Fourier transforms.However, fixed stencil interpolation of second or higher order accuracy is necessarily oscillatorynear a discontinuity. It often leads to numerical instabilities in nonlinear problems containingdiscontinuities.

ENO and WENO schemes are high order accurate finite difference schemes designed forproblems with piecewise smooth solutions containing discontinuities. The key idea is that atthe approximation level, a nonlinear adaptive procedure is used to automatically choose the

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CHAPTER 1. INTRODUCTION 2

locally smoothest stencil, hence avoid crossing discontinuities in the interpolation procedureas much as possible. ENO and WENO schemes have been quite successful in applications,especially for problems containing both shocks and complicated smooth solution structures,such as compressible turbulence simulations and aeroacoustics.

Since the publication of the original paper of Harten, Engquist, Osher and Chakravarthy[11], based on this pioneer work, many reserachers have been improving the methodology andexpanding the area of its applications. ENO schemes are based on point values and Runge-Kutta time discretizations, which can save computational costs significantly for multi-spacedimensions. Later biasing in the stencil choosing process to enhance stability and accuracywere developed. Weighted ENO (WENO) schemes were developed, using a convex combinationof all candidate stencils instead of just one as in the original ENO. ENO and WENO schemescan solve most of the problems of the type in whcih solutions contain both strong shocks andrich smooth region structures. Lower order methods usually have difficulties for such problemsand it is thus attractive and efficient to use high order stable methods such as ENO and WENOto handle them.

The prevalent convergence result for linear conservation laws is based on the Lax Equiva-lence Theorem. Lax Equivalence Theorem equivelates the convergence of a consistent schemeto its stability. In the original paper of G. Jiang and C.-W. Shu [16], for the nonlinear case,there is a proof that for nonlinear conservation laws with smooth solutions, WENO schemesare convergent. This convergence proof is based on the result of Gilbert Strang’s paper [12]and [13], where the Lax stability concept was replaced by the stability of the first variationof the numerical scheme. In this case we refer to [12] for the stability of the first variation.Strang’s theorem (Theorem 1 in [12]) states that, for a conservation law whose flux functionand solution have enough continuous derivatives, a smooth, consistent scheme is convergentif its first variation is l2-stable. The l2-stability of the first variation is hence proved in thepaper of G. Jiang and C.-W. Shu [16]. The convergence result for WENO scheme is actuallya generalization to Lax Equivalence Theorem, see [25], in the nonliear case.

So far, there is no corresponding convergence result for ENO schemes. In the simulationexamples below in Chapter 3, we can see that the derivatives of a numerical solution computedby ENO scheme and its flux do not necessarily behave well even on the continuous part ofa solution. Because of the frequent switching of forward and backward stencil choices, onecannot prove the l2-stability of an ENO scheme’s first variation, and thus the convergenceresult is hard to get within the Strang’s Theorem framework.

Traditionally, when we deal with the error analysis of time-dependent PDEs, we apply thesmoothness of the original solution and the error propagation property of the numerical scheme,as is suggested in the Lax Equivalence Theorem. Since it is extremely hard to prove any errorpropagation property for the complex numerical schemes applied to nonlinear conservationlaws, we alternatively deal with the error analysis of PDEs based on the error propagationproperty of the nonlinear conservation laws’ entropy solutions and the smoothness of thenumerical solutions. We are aiming to use the concept of numerical smoothness to deal withthe error analysis of ENO and WENO schemes, in the similar way where the discontinousGalerkin were dealt with in [26].Normally, numerical solutions consist of non-smooth functionsand discrete point values. However it should resemble the smoothness properties of the originalsolutions. Such resemblence can be represented through some properly defined smoothnessindicators and then applied to error analysis. Our idea on numerical smoothness in the error

CHAPTER 1. INTRODUCTION 3

analysis of hyperbolic conservation laws is a migration of the idea of using numerical smoothingin the error analysis of parabolic equations. See [27] and [28]. The L1-contraction betweenthe entropy solutions is ideal for error propagation analysis in scalar conservation laws. Ithas become more and more evident [26][27][28] that there is also huge advantages to use thesmoothness of numerical solutions for local error analyis.

The error analysis of this paper is based on the way we split the global error. The globalerror can be split into three parts: the propagation of the global error, the local spatialdiscretization error and the local temporal discretization error. Based on this error splitting,we are aiming to design some smoothness indicators to establish the smoothness of the ENOor WENO solutions.

This thesis presents the definitions and numerical experiments of some numerical smooth-ness indicators for the ENO and WENO schemes. We mainly focus on the spatial smoothnessin this work, and we will work on the temporal analysis as the next step following the rou-tine given in [26]. We focus on how to estimate the smoothness of the numerical solutionsof ENO or WENO schemes. The indicators designed here should deliver the smoothness andnonsmoothness information for both continuous solutions and discontinuous solutions. Also,the indicators should tell the smoothness maintenance behavior of different schemes. Basedon the numerical experiments, we are going to analyze the error for continuous solutions anddiscontinuous solutions in the future.

Chapter 2

ENO and WENO

2.1 Reconstruction and Approximation in 1D

In this section we concentrate on the problems of interpolation and approximation in one spacedimension. Given a gird

a = x 12< x 3

2< · · · < xN− 1

2< xN+ 1

2= b (2.1.1)

We define cells, cell centers, and cell sizes by

Ii ≡ [xi− 12, xi− 1

2], xi ≡

1

2(xi− 1

2+ xi+ 1

2),

∆xi ≡ xi+ 12− xi− 1

2, i = 1, 2, · · · , N. (2.1.2)

Denote the maximum cell size by∆x ≡ max

1≤i≤N∆xi. (2.1.3)

We will solve hyperbolic conservation laws using cell averages.Given the cell averages of a function v(x):

vi ≡1

∆xi

∫ xi+1

2

xi− 1

2

, i = 1, 2, · · · , N, (2.1.4)

find a polynomial pi(x), of degree at most k − 1, for each cell Ii, such that it is a k-th orderaccurate approximation to the function v(x) inside Ii:

pi = v(x) +O(∆xk), x ∈ Ii, i = 1, 2, · · · , N. (2.1.5)

In particular, this gives approximations to the function v(x) at the cell boundaries

v−i+ 1

2

= pi(xi+ 12), v+

i− 12

= pi(xi− 12), i = 1, 2, · · · , N. (2.1.6)

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CHAPTER 2. ENO AND WENO 5

which are k-th order accurate:

v−i+ 1

2

= v(xi +1

2) +O(∆xk),

v−i− 1

2

= v(xi −1

2) +O(∆xk), i = 1, 2, · · · , N. (2.1.7)

The polynimial pi(x) can be replaced by other simple functions, such as trigonometric poly-nomials. I will not discuss boundary conditions in this section. I thus assume that vi is alsoavailable for i ≤ 0 and i > N if needed.

Given the location Ii and the order of accuracy k, we first choose a “stencil”, based on rcells to the left, s cells to the right, and Ii itself if r, s ≥ 0, with r + s+ 1 = k:

S(i) ≡ Ii−r, · · · , Ii+s. (2.1.8)

There is a unique polynomial of degree at most k − 1 = r + s, denoted by p(x), whose cellaverage in each of the cells in S(i) agrees with that of v(x):

1

∆xj

∫ xj− 1

2

xj+1

2

p(ξ)dξ = vj , j = i− r, · · · , i+ s. (2.1.9)

This polynomial p(x) is the k-th order approximation I am looking for, as long as the functionv(x) is smooth in the region covered by the stencil S(i).

We also need the approximations to the values of v(x) at the cell boundaries. Since themappings from the given cell averages vj in the stencil S(i) to the values v−

i+ 12

and v+i− 1

2

are

linear, there exist constants crj and crj , which depend on the left shift of the stencil r of thestencil S(i), on the order of accuracy k, and on the cell sizes ∆xj in the stencil Si, but not onthe function itself, such that

v−i+ 1

2

=k−1∑j=0

crjvi−r+j , v+i− 1

2

=k−1∑j=0

crjvi−r+j . (2.1.10)

Let’s denote that the difference between the values with superscripts ± at the same locationxi+ 1

2is due to the possibility of different stencils for cell Ii and for cell Ii+1. If I identify the left

shift r not with the cell Ii but with the point of reconstruction xi+ 12, i.e. using the stencil to

apporximate xi+ 12, then I can drop the superscripts ± and also eliminate the need to consider

crj , as it is clear thatcrj = cr−1,j .

The constants crjare obtained from the Lagrange form of the interpolation polynomial.See [31].

2.2 ENO and WENO Approximations in 1D

For solving hyperbolic conservation laws, we are interested in the class of piecewise smoothfunctions. A piecewise smooth function v(x) is smooth, in other words, it has as many deriva-tives as the schemes calls for, except for at finitely many isolated points. At these points, v(x)


and its derivatives are assumed to have finite left and right limits. Such functions are ”generic”for solutions to hyperbolic conservation laws.ENO approximation.

The basic idea of ENO approximation is to avoid including the discontinuous cell in the stencil,if possible. To achieve this effect, we need to look at the Newton formulation of the interpo-lation polynomial. We first review the definition of the Newton divided differences. The 0-thdegree divided differences of the function V (x) are defined by:

V [xi− 12] ≡ V (xi− 1

2); (2.2.1)

where

V (x) ≡∫ x

−∞v(ξ)dξ. (2.2.2)

Clearly, V (xi+ 12) can be expressed by the cell averages of v(x):

V (xi+ 12) =

i∑j=−∞

∫ xj+1

2

xj− 1

2

v(ξ)dξ =

i∑j=−∞

vj∆xj . (2.2.3)

In general, the j-th degree divided differences, for j ≥ 1, are defined inductively by

V [xi− 12, · · · , xi+j− 1

2] ≡

V [xi+ 12, · · · , xi+j− 1

2]− V [xi− 1

2, · · · , xi+j− 3

2]

xi+j− 12− xi− 1

2

. (2.2.4)

Similarly, the divided differences of the cell averages v are defined by

v[xi] ≡ vi; (2.2.5)

and in general

v[xi, · · · , xi+j ] ≡v[xi+1, · · · , xi+j ]− v[xi, · · · , xi+j−1]

xi+j − xi(2.2.6)

Note that,

V [xi− 12, xi+ 1

2] =

V (xi+ 12)− V (xi− 1

2)

xi+ 12− xi− 1

2

= vi, (2.2.7)

i.e. the 0-th degree divided differences of v are the first degree divided differences of V (x).The Newton form of the k-th degree interpolation polynomial P (x), which interpolates V (x)at the k + 1 points, can be expressed using the divided differences by

P (x) =

k∑j=0

V [xi−r− 12, · · · , xi−r+j− 1

2]

j−1∏m=0

(x− xi−r+m− 12). (2.2.8)

Take the derivative of P (x) to get p(x)

p(x) =

k∑j=1

V [xi−r− 12, · · · , xi−r+j− 1

2]

j−1∑m=0

j−1∏l=0l =m

(x− xi−r+l− 12). (2.2.9)


Notice that only first and higher degree divided differences of V (x) appear in p(x). Hence,we can express p(x) completely by the divided differences of v, without any need to referenceV (x).

To describe the ENO idea, suppose the job is to find a stencil of k + 1 consecutive points,which must include xi− 1

2and xi+ 1

2, such that V (x) is“the smoothest” in this stencil comparing

with other possible stencils. We perform this job by breaking it into steps, in each step weonly add one point to the stencil. Thus, we start with the two point stencil

S2(i) = xi− 12, xi+ 1

2, (2.2.10)

where we have used S to denote a stencil for the primitive function V. Notice that the stencil Sfor V has a corresponding stencil S for v, for example S2(i) corresponds to a single cell stencil

S(i) = Ii

for v. The linear interpolation on the stencil S2(i) can be written in the Newton form as

P 1(x) = V [xi− 12] + V [xi− 1

2, xi+ 1

2](x− xi− 1

2).

At the next step, we have only two choices to expand the stencil by adding one point: we caneither add the left neighbor xi− 3

2, resulting in the following quadratic interpolation

R(x) = P 1(x) + V [xi− 32, xi− 1

2, xi+ 1

2](x− xi− 1

2)(x− xi+ 1

2), (2.2.11)

or add the right neighbor xi+ 32, resulting in the following quadratic interpolation

S(x) = P 1(x) + V [xi− 12, xi+ 1

2, xi+ 3

2](x− xi− 1

2)(x− xi+ 1

2), (2.2.12)

Note that the deviations from P 1(x) are the same function

(x− xi− 12)(x− xi+ 1

2)

multiplied by two different constants

V [xi− 32, xi− 1

2, xi+ 1

2]

andV [xi− 1

2, xi+ 1

2, xi+ 3

2].

These two constants are the two second degree divided differences of V (x) in two differentstencils. A smaller divided difference implies the function is “smoother” in that stencil. Wethus decide upon which point to add to the stencil, by comparing the two relevant divideddifferences, and picking the one with a smaller absolute value. Thus, if

|V [xi− 32, xi− 1

2, xi+ 1

2]| < |V [xi− 1

2, xi+ 1

2, xi+ 3

2]|, (2.2.13)

we will take the 3 point stencil as

S3(i) = xi− 32, xi− 1

2, xi+ 1

2;


otherwise, we will takeS3(i) = xi− 1

2, xi+ 1

2, xi+ 3

2.

This procedure can be continued, with one point added to the stentil at each step, accordingto the smalller of the absolute values of the two relevant divided differences, until the desirednumber of points in the stencil is reached. Note that, for the uniform grid case ∆xi = ∆x, thereis no need to compute the divided differences. We should use undivided differences instead:

V < xi− 12, xi+ 1

2>= V [xi− 1

2, xi+ 1

2] = vi (2.2.14)

and

V < xi− 12, · · · , xi+j+ 1

2>≡V < xi+ 1

2, · · · , xi+j+ 1

2>

−V < xi− 12, · · · , xi+j− 1

2>, j ≥ 1 (2.2.15)

The Newton interpolation formulae should also be adjusted accordingly. this both saves com-putational time and reduces round-off effects.Procedure of 1D ENO reconstruction

Given the cell averages vi of a function v(x), we obtain a piecewise polynomial reconstruc-tion, of degree at most k − 1, using ENO, in the following way:1. Compute the divided differences of the primitive funcion V (x), for degrees 1 to k, usingv. If the grid is uniform ∆xi = ∆x, at this stage, undivided differences should be computedinstead.2. In cell Ii, start with a two point stencil

S2(i) = xi− 12, xi+ 1

2

for V (x), which is equivalent to a one point stencil,

S1(i) = Ii

for v.3.For l = 2, · · · , k, assuming

Sl(i) = xj+ 12, · · · , xj+l− 1

2

is known, add one of the two neighboring points, xj− 12or xj+l+ 1

2, to the stencil, following the

ENO procedure:If

|V [xj− 12, · · · , xj+l− 1

2]| < |V [xj+ 1

2, · · · , xj+l+ 1

2]|,

add xj− 12to the stencil Sl(i) to obtain

Sl+1(i) = xj− 12, · · · , xj+l− 1

2

Otherwise, add xj+l+ 12to the stencil Sl(i) to obtain

Sl+1(i) = xj+ 12, · · · , xj+l+ 1

2.


4. Use the Lagrange form or the Newton form to obtain pi(x),which is a polynomial of degreeat most k − 1 in Ii.

When the stencil is known, it is more convenient to use

v−i+ 1

2

=

k−1∑j=0


2

=

k−1∑j=0

crjvi−r+j .

with crj provided in the table 2.1 of [31], for a uniform grid.WENO approximation

WENO is based on ENO, of course, for simplicity of presentation, we assume the grid isuniform, i.e. ∆xi = ∆x.

As we can see, ENO reconstruction is uniformly high order accurate right up to the discon-tinuity. It achieves this effect by adaptively choosing the stencil based on the absolute valuesof divided differences. However, one could make the following remarks about ENO reconstruc-tion, indicating rooms for improvements:1. The stencil might change even by a round-off error perturbation near zeroes of the solutionand its derivatives. That is, when both sides are near 0, a small change at the round off levelwould change the direction of the inequality and hence the stencil. In smooth regions, this“free adaptation” of stencil is clearly not necessary. Moreover, this may cause loss of accuracywhen applied to a hyperbolic PDE.2. The resulting numerical flux is not smooth, as the stencil pattern may change at neighboringpoints.3. In the stencil choosing process, k candidate stencils are considered, covering 2k−1 cells, butonly one of the stencils is actually used in forming the reconstruction, resulting in k-th orderaccuracy. If all the 2k − 1 cells in the potential stencils are used, one could get (2k − 1)-thorder accuracy in smooth regions.4. ENO stencil choosing procedure involves many logical “if” structures, or equivalent mathe-matical formulae, which are not very efficient.

There have been attempts in the literature to remedy the first problem, the “free adapta-tion” of stencils. In [9] and [21], the following biasing strategy was proposed. One first identitya “preferred” stencil

Spref (i) = xi−r+ 12, · · · , xi−r+k+ 1

2, (2.2.16)

whcih might be central or one-point upwind. One then replaces

|V [xj− 12, · · · , xj+l− 1

2]| < |V [xj+ 1

2, · · · , xj+l+ 1

2]|, (2.2.17)

by|V [xj− 1

2, · · · , xj+l− 1

2]| < b|V [xj+ 1

2, · · · , xj+l+ 1

2]|, (2.2.18)

ifxj+ 1

2> xi−r+ 1

2

i.e. if the left-most point xj+ 12has not reached the left-most point xi−r+ 1

2of the preferred

stencil Spref (i) yet;


otherwise, ifxj+ 1

2≤ xi−r+ 1

2

one replaces|V [xj− 1

2, · · · , xj+l− 1

2]| < |V [xj+ 1

2, · · · , xj+l+ 1

2]|, (2.2.19)

byb|V [xj− 1

2, · · · , xj+l− 1

2]| < |V [xj+ 1

2, · · · , xj+l+ 1

2]|. (2.2.20)

Here, b > 1 is the so-called biasing parameter. Analysis in [18] indicates a good choice of theparameter b = 2. The philosophy is to stay as close as possible to the preferred stencil, unlessthe alternative candidate is, roughly speaking, a factor b > 1 better in smoothness.

WENO is a more recent attempt to improve upon ENO in these four points. The basic ideais the following: instead of using only one of the candidate stencils to form the reconstruction,one uses a convex combination of all of them. To be more precise, suppose the k candidatestencils

Sr(i) = xi−r, · · · , xi−r+k−1, r = 0, · · · , k − 1 (2.2.21)

produce k different reconstructions to the value vi+ 12,

v(r)

i+ 12

=

k−1∑j=0

crjvi−r+j , r = 0, · · · , k − 1, (2.2.22)

WENO reconstruction would take a convex combination of all v(r)

i+ 12

as a new approximation

to the cell boundary value v(xi+ 12):

vi+ 12=

k−1∑r=0

ωrv(r)

i+ 12

. (2.2.23)

Apparently, the key to the success of WENO would be the choice of the weights ωr. We require

ωr ≥ 0,k−1∑r=0

ωr = 1 (2.2.24)

for stability and consistency.If the function v(x) is smooth in all of the candidate stencils, there are constants dr such

that

vi+ 12=

k−1∑r=0

drv(r)

i+ 12

= v(xi+ 12) +O(∆x2k−1). (2.2.25)

For example, dr for 1 ≤ k ≤ 3 are given by

d0 = 1, k = 1;

d0 =2

3, d1 =

1

3, k = 2;

d0 =2

10, d1 =

3

5, d2 =

1

10, k = 3.


We can see that dr is always positive and, due to consistency,

k−1∑r=0

dr = 1. (2.2.26)

When the function v(x) has a discontinuity in one or more of the stencils, we would hope thecorresponding weights ωr to be essentially 0, to emulate the successful ENO idea.

Another consideration is that the weights should be smooth functions of the cell averagesinvolved.

Finally, we would like to have weights which are computationally efficient. Thus, polyno-mials or rational functions are preferred over exponential type functions.

All these considerations lead to the following form of weights:

ωr =αr∑k−1s=0 αs

(2.2.27)

with

αr =dr

(ϵ+ βr)2. (2.2.28)

Here ϵ > 0 is introduced to avoid the denominator to become 0. We take ϵ = 10−6 in all ournumerical tests. βr are the so-called ”smooth indicators” of the stencil Sr(i): if the functionv(X) is smooth in the stencil Sr(i).

As we have seen, the ENO reconstruction procedure chooses the ”smoothest” stencil bycomparing a hierachy of divided or undivided differences. This is because these differences canbe used to measure the smoothness of the function on a stencil.

We define

βr =k−1∑l=1

∫ xi+1

2

xi− 1

2

∆x2l−1

(∂lpr(x)

∂lx

)2

dx. (2.2.29)

When k=2,

β0 = (vi+1 − vi)2,

β1 = (vi − vi−1)2. (2.2.30)

When k=3,

β0 =13

12(vi − 2vi+1 + vi+2)

2 +1

4(3vi − 4vi+1 + vi+2)

2,

β0 =13

12(vi−1 − 2vi + vi+1)

2 +1

4(vi−1 − vi+1)

2,

β0 =13

12(vi−2 − 2vi−1 + vi)

2 +1

4(vi−2 − 4vi−1 + 3vi)

2, (2.2.31)

Procedure of 1D WENO reconstructionGiven the cell averages vi of a function v(x), for each cell Ii, we obtain upwind biased(2k − 1)-th order appriximations to the function v(x) at the cell boundaries, denoted by v+

i− 12


and v−i+ 1

2

, in the following way:

1. Obtain the k reconstructed values v(r)

i+ 12

, of k-th order accuracy, based on the stencils

Sr(i) = xi−r, · · · , xi−r+k−1, r = 0, · · · , k − 1;

Also obtain the k reconstructed values v(r)

i− 12

, of k-th order accuracy, using

v−i+ 1

2

=

k−1∑j=0


2

=

k−1∑j=0

crjvi−r+j .

again based on the stencils

Sr(i) = xi−r, · · · , xi−r+k−1, r = 0, · · · , k − 1;

2. Find the constants dr and dr, such that

vi+ 12=

k−1∑r=0

drv(r)

i+ 12

= v(xi+ 12) +O(∆x2k−1)

and

vi− 12=

k−1∑r=0

drv(r)

i− 12

= v(xi− 12) +O(∆x2k−1)

are valid. By symmetry,dr = dk−1−r.

3. Find the smooth indicators βr, for all r = 0, · · · , k − 1. Explicit formulae for k = 2 andk = 3 are given in (2.42).4. Form the weights ωr and ωr using


αr =dr

(ϵ+ βr)2

and


αr =dr

(ϵ+ βr)2,

r = 0, · · · , k − 1.

5. Find the (2k − 1)-th order reconstruction

v−i+ 1

2

=k−1∑r=0

ωrv(r)

i+ 12

,

v+i− 1

2

=

k−1∑r=0

ωrv(r)

i− 12

. (2.2.32)


We can obtain weights for higher orders of k (corresponding to seventh and higher order WENOschemes) using the same recipe. However, these schemes of seventh and higher order have tobeen extensively tested yet.

2.3 ENO and WENO Schemes for 1D Conservation Laws

In this section, we will describe the ENO and WENO schemes for 1D conservation laws:

ut(x, t) + fx(u(x, t)) = 0 (2.3.1)

equipped with suitable initial and boundary conditions. We will concentrate on the discus-sion of spatial discretization first, leaving the time variable t continuous (the method-of-lineappoach), then apply Runge-Kutta to the time discretizations.

The computational domain is a ≤ x ≤ b. Given a grid

a = x 12< x 3

2< · · · < xN− 1

2< xN+ 1

2= b,

we do not consider boundary conditions here. Assume that the values of the numerical solutionare also available outside the computaional domain whenever they are needed. This would bethe case for periodic or compactly supported problems.Finite volume formulation in the scalar case.

For finite volume schemes, or schemes based on cell averages, we do not solve

ut(x, t) + fx(u(x, t)) = 0

directly, but its integrated version. We integrated it over the interval Ii to obtain

du(xi, t)

dt= − 1

∆xi

(f(u(xi+ 1

2, t)− f(u(xi− 1

2, t)

)(2.3.2)

where

u(xi, t) ≡1

∆xi

∫ xi+1

2

xi− 1

2

u(ξ, t)dξ (2.3.3)

is the cell average. We approximate

du(xi, t)

dt= − 1

∆xi

(f(u(xi+ 1

2, t)− f(u(xi− 1

2, t)

)by the following conservative scheme

dui(t)

dt= − 1

∆xi

(fi+ 1

2− fi− 1

2

), (2.3.4)

where u(t) is the numerical approximation to the cell average u(xi, t), and the numerical fluxfi+ 1

2is defined by

fi+ 12= h

(u−i+ 1

2

, u−i+ 1

2

)(2.3.5)


with the values u±i+ 1

2

obtained by the ENO reconstruction, or by the WENO reconstruction.

The two argument function h is a monotone flux. It satisfies:1. h(a, b) is a Lipschitz continuous function in both arguments;2. h(a, b) is a nondecreading function in a and a nonincreasing function in b.3. h(a, b) is consistant with the physical flux f , that is, h(a, b) = f(a).

Examples of monotone fluxes include:1. Godunov flux:

h(a, b) =

mina≤u≤b f(u) if a ≤ bmaxb≤u≤a f(u) if a > b

. (2.3.6)

2. Engquist-Osher flux:

h(a, b) =

∫ a

0max(f

′(u), 0)du+

∫ b

0min(f

′(u), 0)du+ f(0). (2.3.7)

3. Lax-Friedrichs flux:

h(a, b) =1

2[f(a) + f(b)− α(b− a)] (2.3.8)

where α = maxu |f′(u)| is a constant. The maximum is taken over the relevant range of u.

When choosing the monotone flux, Godunov flux behaves much better for the first orderscheme, while the Lax-Friedrichs flux is simpler and less expensive. We will test these twofluxes in the next section.

Up to now, we have only considered spatial discretizations, leaving the time variable con-tinuous. Now we will consider the issue of time discretization. We will use the method of linesapproach: TVD Runge-Kutta methods.

A class of TVD (total vatiation diminishing) high order Runge-Kutta methods is developedin [22] and further in [10]. These Runge-Kutta methods are used to solve a system of initialvalue problems of ODEs written as:

ut = L(u), (2.3.9)

resulting for a method of lines spatial approximation to a PDE such as:

ut = −f(u)x. (2.3.10)

Clearly, L(u) here is an approximation (e.g. ENO or WENO approximaition above), to thederivative −f(u)x in the PDE.Assume that a first order Euler forward time stepping:

ucn+1 = ucn +∆tL(ucn) (2.3.11)

is stable in a certain norm:∥ucn+1∥ ≤ ∥ucn∥ (2.3.12)

under a suitable restriction on ∆t:∆t ≤ ∆t1, (2.3.13)


then we look for higher order in time Runge-Kutta methods such that the same stability resultholds, under a perhaps different restriction on ∆t:

∆t ≤ c∆t1 (2.3.14)

where c is termed the CFL coefficient for the high order time discretization.The stability condition

∥ucn+1∥ ≤ ∥ucn∥

for the first order Euler forward in time

ucn+1 = ucn +∆tL(ucn)

is easy to obtain in many cases.The TVD high order time discretization defined above maintains staility in whatever norm,

of the Euler forward first order time stepping, for the high order time distretization, under thetimestep restriction

∆t ≤ c∆t1.

A general Runge-Kutta method forut = L(u)

is written in the form:

u(i) =i−1∑k=0

(αiku

(k) +∆tβikL(u(k))

), i = 1, · · · ,m

u(0) = un,

u(m) = un+1. (2.3.15)

Cleaerly, if all the coefficients are nonnegaive αik ≥ 0, βik ≥ 0, then the general Runge-Kutta method is just a convex combination of the Euler forward operators, with ∆t replacedby βik

αik∆t, since by consistency

∑i−1k=0 αik = 1. The method used in our tests is the optimal

third order TVD Runge-Kutta method:

u(1) = un +∆tL(un)

u(2) =3

4un +

1

4u(1) +

1

4∆tL(u(1))

ucn+1 =1

3un +

2

3u(2) +

2

3∆tL(u(2)). (2.3.16)

Chapter 3

Numerical Smoothness

3.1 Error propagation

The idea of numerical smoothness in the error analysis of hyperbolic conservation laws is amigration of the idea of using numerical smoothing in the error analysis of parabolic equations.The main idea is to estimate error by using the error propagation property of a PDE and thesmoothness of a numerical solution. This is an alternative of the most popular Lax EquivalenceTheorem, where the error propagation property of a numerical scheme and the smoothness ofa PDE solution are used. For the scalar conservation laws, there is the L1-contraction betweenthe PDE solutions, which is ideal for error propagation analysis.

ucn

u(tn)ucn+1

uh(tn+1)

u(tn+1)

u(tn+1)

-

6

tn tn+1

As in the graph, in order to analyze the global error u(tn+1) − ucn+1, we introduce two auxil-iary solutions. Let u(t) be the entropy solution of the PDE evolving out of ucn. Let uh(t) bethe semi-discrete solution (solution of (2.3.4)) evolving out of ucn. With these two auxiliarysolutions, we split u(tn+1)− ucn+1 into (3.1)

16

CHAPTER 3. NUMERICAL SMOOTHNESS 17

∥u(tn+1)− ucn+1∥ ≤ ∥u(tn+1)− u(tn+1)∥+ ∥u(tn+1)− uh(tn+1)∥+ ∥uh(tn+1)− ucn+1∥. (3.1.1)

The first part u(tn+1)− u(tn+1) is the propagation of the global error u(tn)− ucn at tn by thePDE. Due to the L1-contraction property of the scalar conservation laws,

∥u(tn+1)− u(tn+1)∥L1(Ω) ≤ ∥u(tn)− ucn∥L1(Ω), (3.1.2)

where Ω = [a, b].The second part of the split error is the local spatial discretization error u(tn+1)−uh(tn+1).

Since ucn lives in the discrete approximation function space, u(tn+τ), for τ ∈ [0,∆t] is certainlynot smooth in the classical sence. However, we can still consider its smoothness in a discretesense and use such smoothness to estimate the error.

The third part of the split error is the local temporal discretization error uh(tn+1)− ucn+1.To estimate this part of the error, we will need the temporal smoothness of uh(tn + τ) forτ ∈ [0,∆t]. Since uh(tn+ τ) is a solution of the ODE (2.3.4), with initial value ucn, we can alsoestablish the needed smoothness.

3.2 Numerical smoothmess and error analysis

In order to define the smoothness of the numerical solution, we want to design some smoothnessindicators. By the result of the paper of D. Rumsey and T. Sun, given by [26], we want todevelop a relation between the error and two smoothness indicators , such that

∥u(tn+1)− ucn+1∥ ≤ ∆xp∆tSp(ucn) + ∆tk+1Tk(u

cn), (3.2.1)

where Sp(ucn) is a spatial smoothness indicator and Tk(u

cn) is a temporal smoothness indicator.

Consequently, the global error at time T with T = N∆t is estimated by

∥u(T )− ucN∥ ≤ T∆xpmaxn

Sp(ucn) + T∆tk max

nTk(u

cn). (3.2.2)

These two smoothness indicators are usually related to:1. the Newton divided differences;2. the derivatives of the recovered numerical solutions inside the cells; and3. the jumps of the derivatives of the recovered numerical solutions on the cell boundaries.

Thus, we will design some indicators to study the behavior of these related factors.

3.3 Smoothness indicators

In this section, we will define some smoothness indicators, which can reflect the smoothnessof the numerical solution of ENO or WENO schemes, and also can detect the location of the


shocks. Our final goal is to develop a set of spatial and temporal smoothness indicators parallelto those in Rumsey and Sun’s paper [26]. However, we will concentrate on the discussion ofspatial smoothness.

Use the procedure of 1D ENO reconstruction or procedure of 1D WENO reconstruction,to obtain the third order reconstructed values u−

i+ 12

and u+i+ 1

2

for all i. Then we will choose

Lax-Friedrichs flux and Godunov flux, respectively, to compute the flux fi+ 12for all i, and then

form the scheme (2.3.4). Next we apply the optimal third order TVD Runge-Kutta method,given by (2.3.16).

We already know the divided difference is defined inductively, given by 2.2.4:

V [xi− 12, · · · , xi+j− 1

2] ≡

V [xi+ 12, · · · , xi+j− 1

2]− V [xi− 1

2, · · · , xi+j− 3

2]

xi+j− 12− xi− 1

2

,

and we need to define the modified Newton’s divided difference:

V1(i) =V (xi+ 1

2)− V (xi− 1

2)

∆x= vi

D1(i) =V1(i+ 1)− V1(i)

∆x(3.3.1)

and

D2(i) =D1(i+ 1)−D1(i)

∆x(3.3.2)

By induction, we have

Dn(i) =Dn−1(i+ 1)−Dn−1(i)

∆x(3.3.3)

where n = 1, 2, · · · , 6.We need to examine these smoothness indicators: 1-st degree modified divided differences and2-nd degree modified divided differences, denoted asD1 andD2. In fact, D1 is closely related tothe first derivative of the numerical solution and D2 is closely related to the second derivativeof the numerical solution, and so on.

We also want to examine the jumps of the first and second derivatives. Using the Newtonform of the third degree interpolation polynomial P (x), given by 2.2.8, and take the derivativeof 2.2.8 to get 2.2.9, where k = 3.

For 1D ENO reconstruction, if we start with the two point stencil

S2(i) = xi− 12, xi+ 1

2, (3.3.4)

the linear interpolation on the stencil S2(i) can be writtenin the Newton form as shown in(2.2). As the next step, we have only two choices to expand the stencil by adding one point:either adding the left neighbor xi− 3

2to get the quadratic interpolation (2.2.11), or adding the

right neighbor xi+ 32to get the quadratic interpolation (2.2.12). Take the derivative of either

(2.2.11) or (2.2.12) to get the first derivative of the numerical solution on both sides of each


cell boundary, i.e. u′−i+ 1

2

and u′+i+ 1

2

, and the second derivative of the numerical solution on both

sides of cell boundary, i.e. u′′−i+ 1

2

and u′′+i+ 1

2

.

For 1D WENO reconstruction, we will always use the central stencil to interpolate thequadratic interpolations.

DenoteJ1(xi+ 1

2) = u′+

i+ 12

− u′−i+ 1

2

(3.3.5)

andJ2(xi+ 1

2) = u′′+

i+ 12

− u′′−i+ 1

2

. (3.3.6)

J1 plays a role similar to V3, the second derivative of the numerical solution of PDE, and J2functions like the third derivative of the numerical solution. These indicators will tell how”smooth” the solution is. However, the difference of these indicators between the neighboringcells is very small(normally 10−6 in our example). Thus, we will put J1 and J2 in the log∆x,to get

log∆x|J1| (3.3.7)

andlog∆x|J2|. (3.3.8)

A certain conbination of these quantities should play the role of the spatial indicator Sp in 3.3.According to (2.3.4), the designed temporal smoothness indicator Tk should be designed in thesame routine given there.

We are going to use D1 through D6 and J1 and J2 as our spatial smoothnessindicators. It is also possible to use high order recovery and compute the jumps of high orderderivatives.

3.4 Simulation

Here, we will study the Burgers’ Equation as the model problem

ut + (u2

2)x = 0, x ∈ [0, 10]

u(x, 0) = 0.2 + 0.5e1+ 1

(x−1)(x−3)

u(0, t) = 0.2. (3.4.1)

The following parameters are used for all of our tests.

∆x = 0.05, ∆t =∆x

10.

First, we apply the 1D ENO scheme, 1D biased ENO scheme (using biasing parameterb = 2) and 1D WENO scheme to (3.4.1), using Lax-Friedichs flux. The solutions are shown in(Fig. 3.1)


0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t=0

t=1

t=2.5

t=4

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t=0

t=1

t=2.5

t=4

t=5

(b) Biased ENO scheme

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t=0

t=1

t=2.5

t=4

t=5

(c) WENO scheme

Figure 3.1: Third order reconstruction, using Lax-Friedichs flux


From (Fig. 3.1), we can see that the wave is travelling to the right. The shock is fullydeveloped at t = 2.5. However, we cannot see too many differences between them. Thus,in order to study the numerical smoothness of ENO, biased ENO and WENO schemes, it isnecessary for us to look into the smoothness indicators we designed above.

Before we look into the smoothness indicators, for comparison between different kinds offlux, we apply the 1D ENO scheme, 1D biased ENO scheme (using biasing parameter b = 2)and 1D WENO scheme to (3.4.1), using Godunov flux. The solutions are shown in (Fig. 3.2).


0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t=0

t=1

t=2.5

t=4

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t=0

t=1

t=2.5

t=4

t=5


0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t=0

t=1

t=2.5

t=4

t=5

(c) WENO scheme

Figure 3.2: Third order reconstruction, using Godunov flux


By comparing the graphs of solutions obtained using Lax-Friedrichs flux and Godunov flux(Fig. 3.1 and Fig. 3.2), we can see that the results of Godunov flux are very similar to thatof Lax-Friedrichs, but with slightly smaller numerical diffusion effect. However, the Godunovflux is relatively more expensive than Lax-Friedrichs flux. Thus, we will use Lax-Friedrichsflux in our future tests.

Next, we will compare D1 of the numerical solutions for ENO, biased ENO and WENOschemes, respectively, at the time t = 0, t = 1, t = 2.5, t = 4, t = 5, as shown in Figures 3.3-3.7.


0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

t=0

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

t=0


0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

t=0

(c) WENO scheme

Figure 3.3: D1 at t = 0


0 1 2 3 4 5 6 7 8 9 10−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

t=1

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

t=1


0 1 2 3 4 5 6 7 8 9 10−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

t=1

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t=2.5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t=2.5


0 1 2 3 4 5 6 7 8 9 10−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t=2.5

(c) WENO scheme

Figure 3.5: D1 at t = 2.5


0 1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

t=4

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

t=4


0 1 2 3 4 5 6 7 8 9 10−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t=4

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t=5


0 1 2 3 4 5 6 7 8 9 10−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

t=5

(c) WENO scheme



3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5−0.2

−0.1

0

0.1

0.2

0.3

0.4

(a) ENO scheme

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5−0.2

−0.1

0

0.1

0.2

0.3

0.4


3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5−0.2

−0.1

0

0.1

0.2

0.3

0.4

(c) WENO scheme

Figure 3.8: zoomed D1 near the shock at t = 4


In Figures 3.3-3.7, we can see that the graphs of D1, i.e. the first derivative of the numericalsolution, are similar to each other. There is a jump at the shock, as well as at the beginningof the wave. However, the jump at the beginning of the wave becomes smoother as the shockdevelops. Then, by zooming in on the part near the shock, given in (Fig. 3.8) we can see thatD1 right before the shock of biased ENO scheme is smoother than that of ENO scheme. Thatis, the biased ENO scheme indeed improves the original ENO scheme.

Next, we will compare D2 of the numerical solutions for ENO, biased ENO and WENOschemes, respectively, at the time t = 0, t = 1, t = 2.5, t = 4, t = 5, as shown in Figures3.9-3.13.


0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

4

6

8

10

t=0

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

4

6

8

10

t=0


0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

4

6

8

10

t=0

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−50

−40

−30

−20

−10

0

10

20

30

40

50

t=1

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−60

−40

−20

0

20

40

60

t=1


0 1 2 3 4 5 6 7 8 9 10−50

−40

−30

−20

−10

0

10

20

30

40

50

t=1

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

80

t=2.5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

80

t=2.5


0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

80

t=2.5

(c) WENO scheme

Figure 3.11: D2 at t = 2.5


0 1 2 3 4 5 6 7 8 9 10−60

−40

−20

0

20

40

60

80

t=4

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−60

−40

−20

0

20

40

60

80

t=4


0 1 2 3 4 5 6 7 8 9 10−60

−40

−20

0

20

40

60

80

t=4

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

t=5


0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

t=5

(c) WENO scheme



1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(a) ENO scheme

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(c) WENO scheme

Figure 3.14: Zoomed D2 near the shock at t = 4


From Figures 3.10-3.13, we can see that the graphs of D2, i.e. the second derivative ofthe numerical solution still behaves similarly: the jump at the beginning of the wave becomessmoother as the shock develops and the jump at the shock becomes steeper and sharper. How-ever, as we zoom in the part near the shock, given by (Fig. 3.14), we can see clearly that forD2 of ENO scheme there is a large area of fluctuation before the shock, and the fluctuationarea is much smaller for both biased ENO and WENO scheme.

Next, we will compare J1 of the numerical solutions for ENO, biased ENO and WENOschemes, respectively, at the time t = 0, t = 1, t = 2.5, t = 4, t = 5, as shown in Figures3.15-3.19.


0 1 2 3 4 5 6 7 8 9 10−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t=0

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

2

t=0


0 1 2 3 4 5 6 7 8 9 10−2

−1.5

−1

−0.5

0

0.5

t=0

(c) WENO scheme

Figure 3.15: J1 at t = 0


0 1 2 3 4 5 6 7 8 9 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t=1

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

t=1


0 1 2 3 4 5 6 7 8 9 10−10

−8

−6

−4

−2

0

2

4

6

8

10

t=1

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−4

−3

−2

−1

0

1

2

3

t=2.5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−4

−3

−2

−1

0

1

2

3

4

5

6

t=2.5


0 1 2 3 4 5 6 7 8 9 10−15

−10

−5

0

5

10

15

t=2.5

(c) WENO scheme

Figure 3.17: J1 at t = 2.5


0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

6

7

8

t=4

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−15

−10

−5

0

5

10

15

20

25

t=4


0 1 2 3 4 5 6 7 8 9 10−15

−10

−5

0

5

10

t=4

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

2

3

4

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

2

3

t=5


0 1 2 3 4 5 6 7 8 9 10−10

−5

0

5

10

15

t=5

(c) WENO scheme



The smoothness indicator J1 acts as the second derivative of the numerical solution. Sincewhen formulating the quadratic interpolation, we always choose the central stencil for WENOscheme instead of choosing different stencils for each cell, the behavior of J1 is much betterthan the others. Our goal is to check the smoothness of our solution, as well as its derivatives,so we need to focus on the smoothness of J1. Since the fluctuation might occur in very smallrange, we need to look closely that this. Thus, we designed the smoothness indicator log∆xJ1.

Next, we will compare log∆xJ1 of the numerical solutions for ENO, biased ENO and WENOschemes, respectively, at the time t = 0, t = 1, t = 2.5, t = 4, t = 5, as shown in Figures 3.20-3.24.


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=0

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=0


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=0

(c) WENO scheme

Figure 3.20: log∆x(J1 +∆x6) at t = 0


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=1

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=1


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=1

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=2.5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=2.5


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

6

t=2.5

(c) WENO scheme

Figure 3.22: log∆x(J1 +∆x6) at t = 2.5


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=4

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

t=4


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=4

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

t=5


0 1 2 3 4 5 6 7 8 9 10−1

0

1

2

3

4

5

6

t=5

(c) WENO scheme



The smoothness indicator log∆xJ1 is designed to show us the smoothness of the secondderivative of the numerical solution. The numerical value of log∆xJ1 can be regarded as theorder of the precision of the first derivative. Near the shock, this value is designed to benegative, and in the smooth part, the larger the value, the smoother the numerical solution.From Figures 3.20-3.24, we will focus on the bottom profile of log∆xJ1 near the shock. We cansee that there is not too much fluctuation for J1 of WENO scheme on the smooth part beforethe shock, and most of the part stays above order 2 until near the shock. However, for J1 ofENO and biased ENO scheme, there is much more fluctuation before the shock, so that somepoints lose the order 1 precision. We will learn later that the fluctuation of ENO and biasedENO scheme is caused by the changing of the stencils for each cell. Obviously, the biased ENOscheme efficiently reduces the unnecessary changing of stencils, so the graph of J1 behaviorsmuch better then that of ENO.

Next, we will compare J2 of the numerical solutions for ENO, biased ENO and WENOschemes, respectively, at the time t = 0, t = 1, t = 2.5, t = 4, t = 5. (Figures 3.25-3.29).


0 1 2 3 4 5 6 7 8 9 10−40

−30

−20

−10

0

10

20

30

40

t=0

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−60

−40

−20

0

20

40

60

t=0


0 1 2 3 4 5 6 7 8 9 10−40

−30

−20

−10

0

10

20

30

40

t=0

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

80

100

t=1

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

80

100

120

t=1


0 1 2 3 4 5 6 7 8 9 10−300

−200

−100

0

100

200

300

400

t=1

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

80

100

120

t=2.5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−80

−60

−40

−20

0

20

40

60

80

100

t=2.5


0 1 2 3 4 5 6 7 8 9 10

−300

−200

−100

0

100

200

300

400

500

t=2.5

(c) WENO scheme

Figure 3.27: J2 at t = 2.5


0 1 2 3 4 5 6 7 8 9 10−200

−100

0

100

200

300

400

t=4

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−300

−200

−100

0

100

200

300

400

500

t=4


0 1 2 3 4 5 6 7 8 9 10

−300

−200

−100

0

100

200

300

400

500

t=4

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−200

−100

0

100

200

300

400

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−150

−100

−50

0

50

100

150

200

250

300

t=5


0 1 2 3 4 5 6 7 8 9 10

−300

−200

−100

0

100

200

300

400

500

t=5

(c) WENO scheme



The smoothness indicator J2 acts as the third derivative of the numerical solution. FromFigures 3.25-3.29, we can see the behavior of J2 of WENO scheme is smoother than the others.For J2 of ENO scheme, there is a large area of fluctuation before the shock, which results inlosing the smoothness of the fouth derivative greatly in the future, and hence undermine thesmoothness of the numerical solution. As in the case of J1, we need to focus on the smoothnessof J2. Since the fluctuation might occur in very small range as well, we need to look closedthat this. Thus, we designed the smoothness indicator log∆x J2.

Next, we will compare log∆x J2 of the numerical solutions for ENO, biased ENO andWENO schemes, respectively, at the time t = 0, t = 1, t = 2.5, t = 4, t = 5, as shown in Figures3.30-3.34.


0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

t=0

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

t=0


0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

t=0

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

t=1

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

t=1


0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

6

t=1

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

t=2.5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

6

t=2.5


0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

4

6

8

t=2.5

(c) WENO scheme

Figure 3.32: log∆x(J2 +∆x6) at t = 2.5


0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

t=4

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−3

−2

−1

0

1

2

3

4

5

t=4


0 1 2 3 4 5 6 7 8 9 10−3

−2

−1

0

1

2

3

4

5

t=4

(c) WENO scheme



0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

6

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4

5

6

t=5


0 1 2 3 4 5 6 7 8 9 10−3

−2

−1

0

1

2

3

4

5

t=5

(c) WENO scheme



The smoothness indicator log∆x J2 is designed to show us the smoothness of the thirdderivative of the numerical solution. The numerical value of log∆x J2 can be regarded as theorder of the precision of the second derivative of the numerical solution. Near the shock, thisvalue is designed to be negative, and in the smooth part, the larger the value, the smootherthe numerical solution. From Figures 3.30-Fig. 3.34, we will focus on the bottom profile oflog∆xJ1 near the shock. We can see that after the shock fully developed, there is not too muchfluctuation for J2 of WENO scheme on the smooth part before the shock, but the order of thesmooth part decreases to between 1 and 2. However, for J2 of ENO and biased ENO scheme,there is much more fluctuation before the shock, so that some points lose the order 1 precision.Admittedly, the graph of J2 of biased ENO scheme has less fluctuation before the shock thenthat of ENO, and hence the solution behaviors more smoothly.

Next, we will compare the shift r of the numerical solutions for ENO and biased ENOschemes, respectively, at the time t = 0, t = 1, t = 2.5, t = 4, t = 5, as shown in Figures3.35-3.39.


0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=0

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=0


Figure 3.35: Cell shifts r at t = 0


0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=1

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=1




0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=2.5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=2.5


Figure 3.37: Cell shifts r at t = 2.5


0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=4

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=2.5




0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=5

(a) ENO scheme

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

t=5



From Figures 3.35-3.39, we can see that wherever the shift of the stencil is changed, therewill be fluctuation in the graph of log∆x J1 and log∆xJ2 of ENO or biased ENO scheme,correspondingly. Thus, we can conclude that the graphs of log∆x J1 and log∆x J2 of ENOscheme have unexpected fluctuation. Since the biased ENO scheme can avoid unnecerssaryshift of stencils, it can reduce the fluctuation effectively and efficiently. In this case, the biasedENO reconstruction behaves better than ENO reconstruction.The smoothness indicator we designed works well to determine the location of shocks and the

difference between all schemes. In fact, the model function we test above is not very smooth atthe beginning of the wave so that we cannot tell how well our smoothness indicators work inthe smooth part of the recovery solution. In order to check how well our smoothness indicatorswork in smooth part of the recovery solution, we need to use a smoother initial condition. We


still test Burgers’ Equation:

ut + (u2

2)x = 0, x ∈ [0, 10]

u(x, 0) = 0.5 + 0.3 sin(2πx

10);

u(x, t) = u(x+ L, t) (3.4.2)

where L is the wave length of one period.We choose ∆x = 0.05,∆t = 0.005 and apply WENO scheme on this problem, using Lax-

Friedrichs flux. After computing time 2000 steps, we get the following solution graph (Fig.3.40):

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1t=0

t=2

t=5

t=8

t=10

Figure 3.40: WENO reconstruction, using Lax-Friedrichs flux

In order to see the performance of our smoothness indicators at smooth part of the recoverysolution, we just need to look at the moment before the shock has been developed. We firstgraph the modified divided differences Di, i = 1, 2, · · · , 6 at the time t=2, when the shock hasyet not been developed. See Figures 3.41)-3.46.


0 1 2 3 4 5 6 7 8 9 10−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Figure 3.41: D1, t=2

0 1 2 3 4 5 6 7 8 9 10−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25



0 1 2 3 4 5 6 7 8 9 10−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5


0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1



0 1 2 3 4 5 6 7 8 9 10−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5


0 1 2 3 4 5 6 7 8 9 10−15

−10

−5

0

5

10

15


We already know that Di is closely related to the i-th derivative of our recovery solution.Since the shock has not been developed yet, all the recovery solutions are smooth at t = 2.From (Fig. 3.41)- (Fig. 3.46) we can see that all the derivatives of D1 to D4 behave well. Evenfor D5 and D6, they are still bounded. In other words, the smoothness indicators can keep atleast 5th order accuracy in error analysis. Thus, it is safe to say our smoothness indicatorsshould work well on the smooth part of the numerical solutions for error analysis.

Chapter 4

Conclusion

In this thesis we studied the numerical solutions of nonlinear conservation laws, using ENOand WENO schemes. From the tests and analysis above, we can draw several conclusions:1. The smoothness indicators we designed in Chapter 3 can efficiently test the location of theshocks;2. This set of smoothness indicators can judge the performance of different schemes;3. This set of smoothness indicators shows that, for smooth solutions, the WENO schememaintains the smoothness of the numerical solution up to order 6. Thus, it should work wellfor future error analysis.

71

Bibliography

[1] A. Harten, High resolution schemes for hyperbolic conservation laws, Journal of Compu-tational Physics, v49 (1983), pp. 357-393.

[2] N. Adams and K. Shariff, A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems, Journal of Computational Physics, v127 (1996), pp. 27-51.

[3] W. Cai and C.-W. Shu, Uniform high-order spectral methods for one- and two-dimensionalEuler equations, Journal of Computational Physics, v104 (1993), pp. 427-443.

[4] J.Casper and H. Atkins, A finite-volume high-order ENO scheme for two dimensionalhyperbolic systems, Journal of Computational Physics, v106 (1993), pp. 62-76.

[5] B. Cockburn, An introduction to the discontinuous Galerkin method for convection-dominated problems, Lecture Notes in Mathematics, v1697 (1998), pp. 151-268

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