Luận án tiến sĩ: Stability and error estimation using entropy functions

Like classical estimates for the computational error, the estimate for the entropy errorreflects an accumulation of local truncation errors.. Therefore, the entropy error estimate can be

A DISSERTATIONSUBMITTED TO THE PROGRAM IN SCIENTIFIC COMPUTING AND

COMPUTATIONAL MATHEMATICSAND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

Melissa D AczonDecember 2006

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ii

got Gerritsen) Principal Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate

in scope and quality as a dissertation for the degree of Doctor of Philosophy

2274-60 Mat(Robert Street)

I certify that I have read this dissertation and that, in my opinion, it is fully adequate

in scope and quality as a dissertation for the degree of Doctor of Philosophy

x2 fee

Cf(Doron Levy)

Approved for the University Committee on Graduate Studies

1H

We have two main objectives in this dissertation: (i) to analyze the splitting mechanism behind

entropy stable finite difference approximations of conservative systems; and (ii) to investigate

global error estimation for nonlinear, hyperbolic partial differential equations

For symmetrizable systems of conservation laws, Olsson used entropy functions to obtainrigorous stability estimates for a family of finite difference schemes that approximate the original

equations [Ols95c] A key element behind the estimates and the resulting schemes is a splittingprocess which uses an entropy function to recast the flux derivative into a skew-symmetric form

Gerritsen applied the splitting concept to the compressible Euler equations [Ger96a]

We studied the splitting process through a parameter which defines both the schemes andthe family of entropy functions that was used by Gerritsen and Olsson for the Euler equations.Our analysis of this parameter enabled us to compare the schemes’ behaviors relative to eachother Our results demonstrate the existence of an optimal value which minimizes errors Bothour theoretical analysis and computational examples show advantages of using the split schemesover their un-split counterparts These benefits include greater accuracy and efficiency for thesolution algorithm, and longer periods of integration We also illustrate the split schemes’ ability

to compute the entropy solution when discontinuities (of different types) exist Our analysis ofboth the stability estimates and entropy errors provides insights into these behaviors

We also derived an error estimate for the entropy function of symmetrizable hyperbolic tems Like classical estimates for the computational error, the estimate for the entropy errorreflects an accumulation of local truncation errors We show that the computational and entropyerrors converge in a similar fashion Therefore, the entropy error estimate can be used to monitorand control global accuracy Since the entropy error estimate utilizes variables already computed

sys-by the discretization kernel, it does not add any substantial cost to the solution algorithm We alsodemonstrate the feasibility of using the entropy error estimate as a monitor function for local gridadaption purposes

iv

Many people have supported, influenced and enriched me over the years It is my pleasure toacknowledge them, especially the following:

e Margot Gerritsen — 1 do not know how I could have done this without you You introduced

me to the HPCC project You answered so many questions I had, even the seeminglyinconsequential, and from afar Then you challenged and pushed my understanding tolevels beyond what I had imagined Words are inadequate to express how truly grateful I

ques-e Pques-ellques-e Olsson and Bques-erti! Gustafsson — Thank you for answques-ering my quques-estions and sharing

your insights with me

e Gene Golub and Walter Murray — You admitted me to a very special program Thank you

for fostering a wonderful learning environment, and for making sure that I finish

e My family, especially my parents — Your belief in me has continued to give me strength andfue] my determination To my friends, former SCCM and Stanford students — do I thank or

curse you for providing all those distractions, from volleyball marathons to cooking

adven-tures? Okay, thank you for helping me keep my sanity All of you are constant remindersthat there is so much more to life than matrices and PDEs

more than I do yours Above all, I treasure your love, for it continues to carry me through

the years, especially when I have doubted my paths

vi

Abstract ivAcknowledgements Y

1 Introduction 11.1 Background and Motivation Q Q ee ]1.2 Stability and Error Estimation Q Q HQ eee eee 21.2.1 Computational Efficiency: Grid Adaption 21.2.2 Reliability: Overall Robustness of Simulation 31.2.3 Grid Quality and Control 2 ee 3

1.3.2 Discretization Method 1 0 Q HQ HH ko 4

2 The Analytic Problem and Numerical Scheme

2.1 The Analytic Problem and

2.2.1 Local AcCUraCy Q Q Q Q Q H es 10

"` 1n nh h TH A4 ee ee 112.2.3 Convergence andErrorControl 2 0 ee ee 132.3 Hyperbolic Conservation Laws 6 es 15

2.4 Well Posedness Through Entropy Functions 1 2 eee ee es 18

vii

2.5 Nonlinear Stability Through Entropy 2 0 2 eee ee ee 26

2.5.1 The Semi-discrete System 002000045 272.5.2 Generalized Stability Estimates 29

2.7.1 Traveling Isentropic Vortex 2 ee ee 342.7.2 Advection of Density Wave (Hump) 352.8 Convergence Tests 2 HQ HQ HQ nạ n kg kh k kia 36

Effect of Splitting Parameter 39

3.1.2 Numerical stability 0 0 ee va 443.2 Numerical Comparisons of Local Accuracy 46

3.4 Evolution of Computational Error 2 ee 52

3.4.2 Advection ofDensityHump 563.43 Initial-Boundary Value Problem ó1

Entropy Conservation and Errors 644.1 Continuous and Semi-discrete Entropy Estimates 64

4.3 Convergence of the Global Entropy 1 6 ee 69

4.3.1 Errors Incurred by not Splitting 69

4.3.2 Boundary Errors 2 ee 70

4.3.4 Convergence of Other Errors 1 ee 76

4.4.2 2D Vortex 2 Q HQ Q HQ Quà kg VN va4.5 Discontinuities and the Entropy Solution 0 2005

4.5.3 Euler System: Contact Discontinuity

Error Estimation

5.2 Cockburn and Gau’s Estimate for Scalar Equations 5.3 Entropy Error Estimate for Systems 2 2 eee

5.3.1 Convergence of Errors: Smooth Case

5.3.2 Convergence of Errors: Non-Smooth

5.5 On the Sharpness of the Entropy Error Estimate 5.5.1 Effectof Norm Used 2 2.2 0 2 ee ee ee

5.5.3 EffectofDIscontinuitles Ặ Q Q 002.0002 000%

5.6.1 Local Comparisons of Errors Ặ 00.0502 eae5.6.2 Comparison of Some Monitor Functions

Conclusion

6.1 Entropy Stability and Splitting 2.0 00.0.0 00000000.6.2 Error Estimation with Entropy Functions 1 6.0 eee eee

The Euler Equations of Gasdynamics

A.2 Linearized Equations and Simultaneous Symmetrization A.2.1 Constant Entropy 6 6 HH HH Quà v.v RaA.2.2 Classical Symmetrization Q Q Q Q ho

A.3 Symmetrization viaEntfODYV Q Q Q Q HH go

1X

1081091101II113114116118119120122123123123

128128130

C Local Truncation and Residual Errors

C.1 Richardson Extrapolation

References

CS SS SS SS

142144

146

2.2

2.3

2.4

2.5

3.1

3.2

3.3

3.4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

2D Vortex, RK-TVD: r;, corresponding to Da,D¿,CFL=01 37

2D Vortex, RK-TVD: 7; corresponding to 2Da,D¿,CEL=035 37

1D, Heun’s: rg corresponding to D4, Dg, CTEFL=0.01 38

2D Vortex, Heun’s: r; corresponding to Da, Dạ, CFLEO.L 38

2D Vortex, Forward Euler: r¿ corresponding to Đa, Dg, CFL=0.1 38

Stopping Times: 1D hump, RK-TVD,Order=6,N=Z2l 45

Stopping Times: 2D Vortex, RK-TVD, Order=6,N=2l 45

Break Times 7,: RK-TVD, Vortex strengthe =6.0 00 51 \|e(T)|| , Comp Times: 7 = 40, & = 1.8, CFL = 0.25, Vortex strenghe=5.0 52 ry(€4(Gj)) and ef: ID Hump, Dạ, N=40,T=0.5 71

ry(En (ô¿)) and ef’: 1D Hump, Ds, W=20,T=05 71

r¿(En (64) and ef’: 2D Vortex, Dạ, N= 40, T=0.5 2 .005, 72 r¿(En (ô;)) and ey: 2D Vortex, Dạ, N= 20, T=0Ú.5 72

Convergence Order: 1D Hump, RK-TVD, D2, T=0.1, CFL=0.1 73

Convergence Order: 2D vortex, Dạ, 7 =0.05, CFL=0.05 73

€,(&;) and ey: ID,pfƒ, Dạ, N=100,T=O1 0.0 73

re(&n): 1D, pƒ, Dạ, T=0.1, CFL=0.1 0.0.00 0.00 ee ee 74 rg: Rarefaction; No = 20, Dp, CFL=0.1,7=0.1 75

r„: Shock; No = 20, Dạ, CFL=0.1,7=05 0.0004 ee eee 75 rg: 1D, RK-TVD-3, CFL=0.05, T=0.02 2 0 ee ee 77 Rarefaction 1: Cr with U, = 0, Up =1, Ax =0.02, CFL=0.8, Dg 95

Rarefaction 2: Cr with U; = —1, Ur = 1, Ax=0.02, CFL=0.8, Dg 95

Shock 1: Cr with Up = 1, Up = —1, Dạ, CFL=05 2.0 0 0404 98 Shock 1: Cr with Up =1, Ur =—1, Dạ,CFL=08 98

Xi

4.19

4.20

4.21

4.22

3.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

Shock 2: Cr with U, = 1, Up =0, D4, CFL=08 2 0 0.00048 101

Shock 3 Cr with U, = 3 Up = 1, Dạ, CFL=0.25 102

Shock 3 Cr with U, = 3 Ủa = 1, Dạ, CFL=05 103

Shock locations for each v„(7): Up = 3, Us=1,Da,T=04 103

Comparison of Cr: py = 10.5, pr = 10.0, D4, dx =0.02, CFL=0.5 105

Convergence Comparison of computational and entropy errors 114

Convergence Comparison of computational and entropy errors 115

Convergence Comparison of computational and entropy errors 115

Convergence Comparison of computational and entropy errors 116

r(&,T);& = —1.8, Dạ, Ax = 0.25, At=0.0625,T =fo+l0 119

r(&,T);&= —10, Dạ, Ax = 0.25, A=0.0625,T=/o+il0 120

r(Ô,T); Dạ, Ax=0.25, At =0.0625,T =fo+l0 121

r(6,T); Dạ, Ac = 0.25, At =0.0625,T =fo+lÔ0 122 r(6,T); Dạ, Ax = 0.02, CFL=0.5,T=t+01 2 20.2 0 eee 122

Xil

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

3.24

3.25

c¡ and ca as functions of &@ 2 QC Q LH Q HH ng Q2 vkv v2 40

Behavior of |p*| as a function ofÔ_ Ặ Q Q Le 42

Spectral Radius Behavior ) Q0 Q Q Q HQ v vở 42

ID: Ì|rs||~ and ||tụ ||~ : pậ, © = 2, various vo, Ö - 47

1D: ||Ts|| and ||tụ || : Độ, various @, Dg ẶẶ 48 1D: ||rs|| and ||ty || 92, De, N = 40, low strengthe 48

1D: ||Ts|| and ||ty | pạ, De, N= 40, highe values 49

2D vortex ||Ts|| and ||tu||.: De, N=40,Lowevalues 50

Order = 4, N= 40, CFL =0.25,T =38.0 0.20.0 0008008 53 Order = 4, N= 40, CFL=0.25,T =40.0 0 0000004 53 Order = 4, N=40, CFL=0.25,T =80.0 0 eee eee 54 Order =6, N= 40, CFL=0.25,T=80.0 2.2.0 000500 | 54 Order = 6, N= 40, CFL =0.25,T=2100 0000.4 55 l|e(t; &) ||, v = (1,0), Order=4, N=40, CFL=0.25, higher |@| 56

l|e(t; &) ||, v = (1,0), Order=4, N=40, CFL=0.25, higher& 57

lle(t;6) ||, v = (1,0), Order=4, N=40, CFL=0.25, lower |lôÊ| 57

l|e(t; &)||, v = (1,0), Order=6, N=40, CFL=0.25, higher& 58

llcứ: &) ||, v = (1,0), Order=6, N=40, CFL=0.25, lower |@| 58

|e(&) ||, v= (1,0), Order=4, N=80, CFL =0.25,lowerT 59

|e(&) ||, v = (1,0), Order=4, N=80, CFL =0.25, higherT 59

lIe(ô)|| v= (1,0), Order=6, N=80, CFL=0.25,lowerT 60

l|e(@&)||, v= (1,0), Order=6, N=80, CFL = 0.25, higherT 60

||e(&)|| 1D hump, T=30; Order=6, CFL =0.1,N=80 61

||@(&) || 2D hump, Order=6, CFL = 0.25,N=40 62

Xi

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

4.19

4.20

4.21

4.22

4.23

4.24

4.25

4.26

4.27

4.28

4.29

4.30

Erp — 1D, RK-TVD, Dạ, N=80,TH=1.0 2.2 eee ee 79

e¡(ô; Ar) /A% - 1D, RK-TVD, De, N=80,T=10 co 80

En, - ID, RK-TVD, Dạ, N=80, T= 1Ú ẶQ ee 80 Erp - RK-TVD, Do, N= 80, T=10.0 2 2 ee Se 81

&4 (6; Ar) /fiỆ — 1D, RK-TVD, Dp N= 80, T=10.0 81

&n, - ID, RK-TVD, Dg, N= 80, T=10.0 1 eee ee eee 82 Erg — 1D, Heun, Dạ, N= 80, TH1.0 2 0 ee 83 e(G; At) /ñỂ — ID, Heun, Dg, N=80, T=1.0 00.00 005 83 En, — 1D, Heun, Ds, N=80, T=1.0 2 eee ee 84 Err — 1D, Heun, Dg, N=80, T=10.0 2 ee ee 84 £5 (6; Ar) /ñỆ - 1D, Heun, Ds, W= 80, T=10.0 co 84 €n, — 1D, Heun, Dạ, W=80, T= 100 2 hs 85 Erg — 2D Vortex, RK-TVD-3, De, N=40,7=10 86

€4 (6; At) /fiỆ - 2D Vortex, RK-TVD-3, Ds, N=40,7=10 86

en, - 2D Vortex, RK-TVD-3, Dg, N=40,T7=l0 co 87 Erp — 2D Vortex, RK-TVD-3, Dạ, N=40,T7=30 87

e¡ (ô; At) /ñỆ - 2D Vortex, RK-TVD-3, Ds, N=40,T7=30 88

En, - 2D Vortex, RK-TVD-3, Dp, N=40, T=30 2.0 ee eee 88 Erg — 2D Vortex, Heun, Dg, N=40,T=10 © 2 ee ee ee 89 €4 (6; At) /fñỆ — 2D Vortex, Heun, Dạ, N=40,T=10 - 89

£n, ~ 2D Vortex, Heun, Ds, N=40, T=10 20 00-2 eee eee eee 90 €n, - 2D Vortex, Heun, De, N= 40, T=20 0.2.0 ee ee eee 90 Erp — 2D Vortex, Heun: Dạ, N=40,7T=430 Ặ ees 90 € (0; Ar) /& — 2D Vortex, Heun, De, N=40,7=30 - 90

Rarefaction 1: U, =0, Up =1, dx =0.02, CFL=0.8, Dy, T=0.32 94

Rarefaction 1: U, = 0,= Up = 1, dx = 0.02, CFL=0.8, Dạ, T=0.64 94

Rarefaction 2: U, = —1, Ur =1, dx= 0.02, Dạ, CFL=0.8, T=0.64 96

Rarefaction 2: U; = —1, Up = 1 dx =0.02, Dạ, CFL=0.8, T=1.28 96

Shock 1: Uz = 1.0, Up = —1.0, Dạ, dx = 0.02, T=0.8,CFL=0.5 98

Shock 1: Ứ¿ = 1.0, Up = —1.0, Dạ, dx = 0.02, T=0.8,CFL=08 99

XIV

4.33 Shock 3: U, =3, Up = 1.0, Da, dx = 0.02, CFL=0.25,T=04 1024.34 Shock 3: U, = 3, Ur = 1.0, Dy, dx = 0.02, CFL=0.5,T=04 1034.35 p_ = 10.5, pr = 10.0, Dy, dx = 0.02, CFL=0.5,7=0.5 105

4.36 (eX (T;6;) —eq(T;hi)), THOS oe ee 106

4.39 ||e(T)||«), T=Ú.Š eee 107

5.1 1D hump, RK-TVD-3,CFL=0.1,ê& = —6, Ds, N=80,T=10 1245.2 1D hump, RK-TVD-3, CFL=0.4, & = —2, Dg, N=80,7=10 124

5.3 Contact Discontinuity, RK-TVD-3, CFL=0.5, & = —10, D4, N= 100, T=0.5 1255.4 Contact Discontinuity, RK-TVD-3, CFL=0.5, & = —1.8, Dạ, N= 100, T=0.5 1255.5 1d hump, RK-TVD-3, CFL=0.4, &@ = —2, Dạ, N=80,T=10 1265.6 Contact Discontinuity, RK-TVD-3, CFL=0.5, & = —1.8, D4, N= 100, 7=0.5 127

1.1 Background and Motivation

An NSF-funded Grand Challenge Project to develop an Earth Observing System (EOS) coupled

with atmospheric and oceanic models motivated the work presented here In this set-up, a based observation system interacts with a computer model which carries out simulations fasterthan real time and also estimates where data need to be collected for sufficient accuracy in the

satellite-predictions [OSFT93] The model requests new information from the observation system, the

system gathers the requested data, and that data is then assimilated back into the computationalmodel This strategy reduces the number of observations made, the storage required for them, andthe bandwidth needed between the observing system and the simulation

A group at Stanford [OSF*93, OSF*99] constructed a laboratory experiment that simulates

flows similar to those of interest in the EOS Project The experiment involved a stratified flow inarotating annular tank which exhibits characteristics of large-scale atmospheric and oceanographicflows For a thorough description and discussion of this set-up, see [Ger96b] The computationalmodeling followed two directions: one involved a parallel implementation of a three-dimensional,unsteady Navier-Stokes code for incompressible flow, with continuity and density-transport equa-tions [OSF*99, Cui99]; the other track involved the construction of stable finite difference meth-ods (FDMs) which allow a robust grid adaption routine and error control

In Zhu’s dissertation, data structures for composite-adaptive grids (CAGs) were developedand implemented in one, two and three dimensions for low-order methods [Zhu96] Compositegrid methods are desirable for: (i) their ability to deal with complicated geometries; and (ii) theirlower administrative costs (relative to those associated with unstructured grids) The adaptive

component, which Zhu based on estimates of the local truncation error, helps to make the solutionstrategy even more efficient by restricting high grid density to limited portions of the domain Zhualso applied stability results by Olsson [Ols95a, Ols95b] to obtain convergence of these methodsfor linear problems Consequently, the global computational error could be estimated and con-

trolled through local means Zhu primarily addressed linear problems and proposeda linearization

approach to handle nonlinear applications

The work of Olsson and others [GO94, 0094, Str94, Ols95c] also lead to the development

of strictly stable finite difference methods that became the primary building block of a solutionstrategy presented in Gerritsen’s dissertation [Ger96a] These methods were applied to the com-pressible Euler equations in one and two dimensions For nonlinear systems of conservation laws,these methods are based on two key elements:

(i) the use of homogeneous entropy functions to split and re-write the flux derivative as a sum

of a formally conservative part and a non-conservative term containing entropy variables;

(ii) the application of high-order spatial discretizations which satisfy a summation-by-parts

refine-solution strategy is an artificial viscosity term which is added only in the vicinity of the shock to

suppress spurious oscillations and which retains the stability property of the SHOEC scheme

1.2 Stability and Error Estimation

As implied above, stability and error estimation play a significant role in the development of

computational models We examine this role in three ways

1.2.1 Computational Efficiency: Grid Adaption

One of the difficulties encountered in direct simulations is the limitation of computing power, both

in speed and memory For many problems of interest, the solution is smooth in a large part of the

domain but develops local structures needing finer resolution Placing a fine mesh over the whole

domain to capture isolated local behavior places tremendous strain on computational resources Amore efficient approach uses a high-order method on a base coarse mesh and employs local gridadaption to refine the mesh where interesting phenomena need to be resolved, i.e it places moregrid points at only these critical regions Local error estimation (through a variant of RichardsonExtrapolation), among other strategies, can be used to locate these regions needing refinement.See [Ver96], [BBSW94] and the references therein for a more comprehensive presentation

1.2.2 Reliability: Overall Robustness of Simulation

Closely related to achieving local resolution is obtaining overall robustness of the simulation.This means having reliable estimates of the global accuracy of the computed solution Ideally, we

would like a quantitative relationship between the local error estimates (which can already be part

of the solution algorithm as a criterion for local grid adaption) and the global error, because then

we can easily achieve global control of accuracy through local control Stability lies at the heart

of the process

Stability theory for linear initial value problems is already well developed The work of Olssonand others [Str94, Ols95a, Ols95b] extended the theory to cover a wide class of initial-boundaryvalue problems and numerical methods As we will see in Chapter 2, stability combined withconsistency leads to convergence in the linear case Stability analysis produces energy estimates

of the computed solution’s growth For linear problems, this automatically leads to estimates of

the global error

Unfortunately for nonlinear problems, the classical framework (stability @ consistency =>

convergence) no longer holds, and new difficulties arise One way to obtain stability and error

es-timates is to linearize the equations and then apply linear stability and convergence theory When

it comes to actual global error estimation, this approach has two major drawbacks First, it canoverestimate the error, which leads to more refinement than necessary Second, the linearizationprocess involves extensive computation of Jacobian matrices and their derivatives which adds

significant cost to the solution algorithm

1.2.3 Grid Quality and Control

As discussed above, a good solution algorithm should use both local and global error estimation

As a by-product of these two processes being carried out in tandem, the algorithm efficiently and

robustly predicts the solution, and also provides insight into where measurements must be taken

to keep the simulation on track with the physical set-up In other words, the algorithm predictsthe computational grid needed for a desired accuracy

1.3 Scope of Dissertation

The purpose of this dissertation is two-fold:

1 Analyze stability and convergence properties of the entropy splitting which were not dressed in the works of Gerritsen and Olsson [Ger96a, GO96]; and, in particular, the effects

ad-of a parameter which controls the splitting

2 Investigate other approaches to nonlinear global error estimation which can be used to trol global accuracy

calculations also fit into this category [OS78] Results for these inviscid equations can be extended

to include the addition of viscous terms Our computations primarily use the Euler equations for

gasdynamics We also use Burger’s equation in some of our investigations

1.3.2 Discretization Method

The choice of a numerical method to solve problems such as the Euler and Navier-Stokes

tions involves two components which are closely connected to each other: discretizing the tions and creating the computational mesh or grid The first process can be roughly divided into

equa-finite difference, equa-finite element and equa-finite volume methods, while the latter process can be roughlydivided into structured and unstructured gridding See [Hir88a] for a cursory overview of theseapproaches and the references therein for more comprehensive treatment The actual application

or specific problem to be solved ultimately dictates the chosen approach In this work, we limitourselves to finite differences and structured grids because of their simplicity

1.4 Overview and Summary of Results

The organization of this dissertation and the locations of our contributions are as follows:

e Chapter 2 presents the analytic problem and its discretization We use Olsson’s framework

of entropy well-posedness and stability to obtain additional energy estimates for the

com-puted solution; and, later, to derive a global error estimate for the entropy of conservativesystems We derive necessary conditions for entropy functions used in the splitting process

e Chapters 3 and 4 analyze the flux splitting process and a free parameter & which definesthe resulting schemes We demonstrate the advantage of the split schemes over their unsplitcounterparts Our results show the existence of an optimal value which minimizes errors

in the computations Chapter 3 focuses on traditional errors, while Chapter 4 uses entropy

related measures Our analysis of the global entropy error provides insight into its sources

and enables a comparison of the different schemes’ behaviors We show that conservation

of the global entropy is necessary for the convergence of other errors When discontinuitiesarise in a problem, we show that the split schemes compute the entropy solution

e Chapter 5 investigates global error estimation We derive an error estimate for the entropy

of conservative systems and use this estimate to prove entropy convergence in smooth plications We show that the entropy error is a viable alternative (to the computational error)for monitoring and controlling global accuracy We also investigate the feasibility of usingthe entropy error estimate for local grid adaption purposes

ap-We conclude in Chapter 6 and discuss future research directions

The Analytic Problem and Numerical Scheme

In this chapter we present the analytic problem and its discretization We first briefly discussthe linear case and introduce general principles which lead to convergence results In the secondpart, we turn to the nonlinear case where we focus on hyperbolic conservation laws We includeOlsson’s [Ols95c] generalization of stability into the nonlinear regime through the use of entropyfunctions We use this framework to derive necessary conditions on entropy functions used forsplitting

2.1 The Analytic Problem and Well-posedness

Throughout this discussion, we consider the initial-boundary value problem:

2 u(x,t) = Lu) +F (30), xeQC%#,0</<7 (2.1)

u(x,0) = u°(x) xEQ (2.2) LPu(x,t) = g(x,t), xeT=0o9,0</<7 (2.3)

where € RK”, L is a linear spatial differential operator, L? is a linear boundary operator, and dQ

is the boundary of Q

Hadamard introduced the concept of well-posedness A well-posed problem is one in which

a unique solution exists and depends continuously on the problem’s given data Existence and

uniqueness are minimal requirements for a reasonable problem The third condition guarantees

that small perturbations, such as measurement errors in the data, do not overly affect the behavior

of the solution The books by Kreiss and Lorenz [KL89] and Gustafsson, Kreiss and Oliger[GKO95] provide comprehensive treatment of the subject We highlight some pertinent results.One approach to proving existence for (2.1 — 2.3) is to use difference approximations becauseshowing the existence of solutions for these approximations is not difficult See [RM67, Joh82]for convergence of these approximations The main results show that for a large class of operators,

C*-smooth data (which is dense in Ly) leads to C”-smooth solutions A priori estimates of the

solution and its derivatives can then be obtained in terms of the given data

For Cauchy problems (pure initial value) and periodic initial-boundary value problems, and

in particular those with constant coefficients, the main tool for analysis is the Fourier tion For problems which involve boundaries and non-periodic boundary data, the energy methodprovides greater flexibility Aside from handling more general boundary conditions, the energymethod of establishing these estimates also allows a unified treatment of many problems: con-

transforma-tinuous and discrete models; linear and nonlinear problems; scalar equations and systems Weillustrate this method with a simple example

The Energy Method: A Model Problem

In (2.1), let Lu = au,, where a > 0 is a constant, Q = [0,1] and u(1,t) = g(t) Define the scalar

product (u,v) = fy u(x)v(x)dx, and its associated norm ||w||? = (u,u) ! Then:

d 2

a Mot = Gur) + (mu) + 2(u,F)

= (u, aux) + (aux, 1) + 2(u, F)

= —(auy,u) + (au”)|Š + (auy,u) + 2(u,f) (Integration by part)s

< au(1,t)? — au(0,t)? + 2||w|| ||F | (Cauchy Inequality)

<au(1,t)? — au(0,t)? + ||w|Ÿ + FI’ (Algebraic Inequality)

‘Note that the inner product and its induced norm are time-dependent

lu(-,t)||2 + af u(0,2) dt < é |Inc.0)Í + af u(t,z)ar + [ IF C9) ay

=# [moc I? +a fear + [rca iret]

The above derivation illustrates the underlying principles of the method First, the energy of the

system was defined (in our model problem, it is the L2-norm), and then differentiated with respect

to time Next, the differential equation is applied, along with several tools (integration by parts,

Cauchy inequality, algebraic inequality) to arrive at an energy estimate This estimate bounds thenorm of the solution in terms of the given data We will see in the next several sections that moremachinery is needed for a parallel treatment of the discrete case and nonlinear problems

We now make a quantified formulation of well-posedness First, we define other norms (for

more general dimensions) in the standard way: |u|* = u*u, where + denotes the conjugate

trans-pose, |ulŠ = ƒalxl24x, Nel 2co.rj = Jo la la axa, and lll! 91) = Jo Sr lul? axe.

Definition 2.1.1 The initial-boundary value problem defined by (2.1 — 2.3) is said to be well posed

if there exists a unique solution which satisfies the estimate

where K and are constants (independent of u°,g,F), t € [0,T].

For linear problems, we can omit the “unique” part in the definition, since estimates of theabove form immediately lead to uniqueness Among the class of problems which have beenshown to be well-posed (for a large enough class of admissible data) are hyperbolic, parabolicand mixed hyperbolic-parabolic types [KL89, GKO95] These results are important because manyflow problems of interest fall into one of these categories Rigorous analysis of these problems

can also be found in [Ols95a, Ols95b]

2.2 Discretization

As we indicated in the introduction, we limit our presentation to finite difference methods LetO¿;,T; and [0,7], be the discretizations of Q, T and [0,7] respectively Let v, be a grid function

defined on Qy, x [0, 7]¿ Without loss of generality and for ease of notation, we discretize equations

(2.1 — 2.3) as

vp(t + At) = On kva(t) + AtF,(t) Qy x (0, TÌy (2.5)

w„(0) = v°, Qn (2.6)

(L3 4)" valt), = #n() Tp x (0, T}x (2.7)

where Qn, denotes the difference operator written in one-step form We use the subscripts h,k

to denote projections of functions onto the corresponding grids and discretizations of operators

on these grids More generally, a difference approximation can be written in the expanded form[GKO95]

and the subscript h again denotes spatial discretization of the operators and data

Before we proceed further, we need to define some (discrete) norms on the grids For ease ofnotation, we assume a uniform spacing, Ax, in the different (spatial) directions

lImOlla, = DY Adojlyny,t)/?

x; củy

ls:lÖ,xo„, = Y A/Av~'gj|g,(x;t;)|JÏx;€T›,f;€|0,]y

x¡ cQ¡,¡€|0,]:

In the above expressions, d is the spatial dimension defined in (2.1) The G;”s are positive weightscorresponding to more general weighted norms In most cases, they are uniformly equal to one.

We see in later discussions that there is a need for different weights

2.2.1 Local Accuracy

If we wish the solution of a numerical scheme to be “close” to the exact solution of the analyticproblem, then the discretization itself must be “close enough” to the original differential equa-

tions The concepts of consistency and order of local accuracy provide concrete measurements

Definition 2.2.1 If uj, is the projection of a smooth solution u(x,t) of (2.1 — 2.3) onto Qy x [0,T]x

then the local truncation error 7); at time t corresponding to (2.5) is defined by

Attn, (t) = un(t + At) — On xun(t) — Atf() €x |0, TÌu (2.11)

Furthermore, the discretization (2.5) is said to be accurate of order (pj, p2) if, for any sufficientlysmooth solution u(x,t),

thk = O(Axf! + Af??), (2.12)

If pị > 0 and p2 > 0, then the approximation is called consistent

This definition measures local discretization errors in terms of the exact solution It is alsopossible to examine these errors in terms of the computed solution

Definition 2.2.2 Let v be a continuous interpolant of the computed solution vụ of the

discretiza-tion (2.5) The local residual rụ ¿ is defined by:

"hk = 2y—L0)-F (2.13)

The local truncation error answers this question: “How well does the analytic solution satisfythe discrete problem?” The residual addresses this point: “How well does the computed solutionsatisfy the analytic problem?” If the solution has enough regularity, then +„„ and r;„ have thesame order of magnitude with respect to the discretization grid sizes At, Ax (see Appendix C for

details)

Higher order methods (meaning those with larger p;) have been shown to be advantageous.See, for example, the work of Kreiss and Oliger [KO72]

2.2.2 Stability

When we numerically solve the analytic problem (2.1 — 2.3), local errors from round-off anddiscretization accumulate, and some form of stability controls this accumulation Stability isessentially the discrete version of well-posedness, and thus provides a way to measure the growth

of the computed solution in terms of the data There are several formulations of stability, but wechoose one which parallels our definition of well-posedness

Definition 2.2.3 The difference approximation (2.5) is said to be stable if the following estimate

holds for its solution vụ:

IIvn(t)Ild, < Ket" (Ini, + IlFalld, <io4y, + ImÊ spa, )› (2.14)

where K' and G are constants (independent of the data), and t € [0,T]x

In general, the constants K’ and œ are not the same as their counterparts in the analytic case

(see Definition 2.1.1) For some problems and discretizations, we can show that K’ = 1 and

œ = œ+ O(Ax) Olsson refers to this property of a discretization as strict stability [Ols95a] Theimportance of strict stability will become more evident when we discuss convergence and errorcontrol

Stability analysis for discretizations of periodic problems most often employs the Fourier

transformation, and is frequently referred to as Von Neumann analysis The process involvesdetermining the symbol associated with the method then showing that it is bounded independent

of the grid size [RM67, GKO95] As in the analytic case, analysis for more general

initial-boundary value problems requires more flexible tools (such as the discrete energy method) andother machinery

Summation by Parts

The model problem illustrates one of the key steps of the energy method: integration by parts.The process takes advantage of the skew symmetry of the differential operator d/dx to eliminateterms involving this operator It is desirable to have a discrete analog of this process

Strand and others [Str94] have generated families of difference operators of arbitrarily

high-order (as approximations for d/dx) which lend themselves to this process These operators have

the form:

Br 0 0

D=|0 De 0

0 0 Br

The skew-symmetric matrix Dc represents the interior stencil, while the matrices B; and Br are

biased boundary operators See Appendix B for some examples Furthermore, D satisfies a

dis-crete version of integration by parts, which we refer to as the summation by parts (SBP) property:

(u, Dv), = —(Du, v)y + MÃ — ub,

where the subscript h denotes a discrete weighted inner product defined by the following:

(1,D(uxv)), = (1,uxDv); + (1,v*Du)p

In the above expressions, uxv = (uA vo, ,uyvy)? denotes the component-wise scalar product of the grid vectors u and v, and 1 = (1, 1)”.

We now discretize the model problem using any difference operator D satisfying SBP:

d 0

Ty =aD+P v(0) = v' (2.15)

Applying the discrete energy method yields

d

which parallels what we had before

Projections for Boundary Conditions

Equation (2.15) is a system of constant coefficient ordinary differential equations (ODEs) with

a unique analytic solution Consequently, that solution may not satisfy the analytic boundarycondition which has not yet been incorporated into the numerical scheme We therefore enforcethe boundary condition:

(1) v= g (2.17)

As long as the initial and boundary data are compatible, the above procedure — solving equations(2.15) and (2.17) — is equivalent to solving the equations

d

3y = PlaDv + F) + (I— Par, v(0) = v°, (2.18)

where P denotes a projection onto the space W = {w: (L)Ïw = 0}, and %, is an auxiliary tion which satisfies (LB )’ = g [Ols95a] With a suitable choice of P, the formulation given by

func-equation (2.18) facilitates the energy method, and thus allows us to obtain stability estimates forthe numerical approximation v (see inequality (2.16)) which incorporates the boundary data For

a thorough discussion, see [Ols95a] We emphasize these points: (i) we use the auxiliary tion (2.18) for analysis only; and (ii) when we compute the solution, we use equations (2.15) and(2.17)

equa-2.2.3 Convergence and Error Control

We define the error by e;,(t) = u;,(t) — v,(t), where uz and v„ denote the exact and approximatesolutions defined on the grid Q, For linear systems, consistency implies

en(t + At) = Lnxen(t) + Attn (t)en(0) = init

(Lin) en(t) = ebary:

Stability then automatically yields a bound on the error:

/ 2 2

Ile„)llỗ, < K'eTM" (a +lltsxlÌ6,xIoa, + lena oe ) (2.19)

If we know the local accuracy, then we also know the global accuracy This is the crux of the

so-called Lax-Richtmyer Equivalence Theorem This fundamental result states that consistencyand stability are necessary and sufficient conditions for a linear difference method to converge

to the solution of a well-posed problem [RM67, GKO95] The derivation of the error equations

above hinges on linearity Hence this equivalence no longer holds in the nonlinear case, and

consequently, the concepts of stability and convergence need to be extended

Olsson’s framework for stability analysis, which includes summation by parts and boundaryprojections, therefore allows convergence results to be established for a large class of linear initial-

boundary value problems Furthermore the rate of convergence, which is governed by the order

of accuracy of D, can be as high as we want Zhu applied Olsson’s stability results to proveconvergence on adaptive grids, and used the strict stability assumption (K’ = 1) in the process

[Zhu96] This is not a a major difficulty If the original estimate has a constant K’ that is equal to

1+ O(Ar), then we can obtain a new estimate whose constant K’ is equal to one We then absorb

the rest into the term e%7 through a modified growth factor a’ The difficulty lies in ensuring that

this new growth factor satisfies o/ = «+ O(Ax) If this assumption does not hold, then the errors

will dominate the solution

Alternatively, we can also derive a bound using the residual:

lle@lla < KeTM (ews + leo lesa loa) (2.20)

where £(-,f) = u(-,t) — v(-,f), and v is a continuous interpolant of the computed solution

If we wish to control the global error, then either estimate (2.19) or (2.20) tells us that wecan do so by controlling the local errors Suppose that strict stability holds Further let ồ be themaximum of the data and local errors: ồ > max((||x||, |lellinit||; ||ellbary) Then

llen(T)lla, S ồ: /e#Tm(O;)7, (2.21)

where (Q,,) is the measure of the computational grid Thus, if e is the desired global accuracy,then the following prescribes the tolerance for the local errors:

€max(||t||,|leinit|l, llevay ||) < ỗ = Tara (2.22)

We can enforce this tolerance by either reducing the mesh size or increasing the discretization’sorder of accuracy The first approach allows flexibility in creating the grid: we can start with afew number of grid points then add more only when necessary, in response to (2.22)

We can approximate the growth factor a with the growth factor of the computed solution vụ,This works well for linear problems The nonlinear case poses new difficulties because stability

and convergence analysis become more complicated One approach to obtaining error estimates

is to linearize the problem However, the growth rates obtained from this process may not be verytight Further, the actual computational costs can be prohibitively high Olsson conjectured that

we can still use the growth rate of the computed solution even in nonlinear cases This approach

is certainly much more cost effective than linearization In the next two sections and in Chapter

5, we discuss extensions of stability, convergence and global error estimation to the nonlinearregime

2.3 Hyperbolic Conservation Laws

We focus our analysis and computation on the following system of time-dependent, first-orderpartial differential equations written in conservative form:

get De rw = 0, x=(x1, ,%¢) €QCR4, +>0 (2.23)

u(x,0) = u°(x), (2.24)

LP u(x,t) = g(x,t) xeT=oQ,:>0 (2.25)

u:(Qx [0,T]) 9", /#:9W" 9", (2.26)

We assume that the system is hyperbolic, which means that any real combination of the flux

Jacobians (3; a¡50)) is diagonalizable with real eigenvalues.

It is well known that the system admits solutions which develop singularities (e.g shocks) infinite time, even when it is given smooth initial data (see e.g [Lax73, Ser99] for basic facts onconservation laws) Such solutions no longer satisfy the PDE in the classical sense, but they can

be understood in terms of the integral form of the conservation law, or in the sense of distributions

Specifically, w is called a weak or generalized solution to (2.23) if for all test functions ộ € C}(Q x

[0,7]), T

[ [leu + (V-6)-/0)|dxár = [ A060)86,004x

Unfortunately, infinitely many weak solutions often exist, which then poses mathematical and

numerical difficulties However, if the system is to model the physical world, then only one ofthese solutions is acceptable, and knowledge of what is being ignored by the equations must be

used to select this unique solution Two mechanisms, both motivated by physical considerations,are frequently used: vanishing viscosity and entropy conditions.

The hyperbolic conservation law can be regarded as an approximation to the following viscous

is associated with the conservative system (2.23) and corresponding flux functions g', it followsfrom (2.27) that [Ols95c, Tad97]

When we are in the smooth regime, we can linearize the equations to study the problem’swell-posedness and the convergence behavior of its discretizations The next several sectionsand chapters show that by employing entropy functions, we can extend the previous concepts

of well-posedness and stability to the nonlinear regime, and the process makes no smoothnessassumptions Therefore in the entropy framework, we can again analyze well-posedness, stability,

convergence and error estimation; and we can do so for a wider class of problems than what the

linearized framework covers

2We will define this more precisely in a subsequent section

2.3.1 Linearized Equations

In smooth parts of the domain, we can linearize the equations to study properties of the solution

and convergence of discretizations Suppose that Ø(x,f) solves the quasi-linear system Ÿ

where F = [u® -Ã'(U)]uỆ + r(U), and r(U) = —U, — Ã!(U)U,, Thus far, we have not neglected

any terms, and so the above equation is exact If we assume that u® is small enough, then we can

drop the quadratic term in F and obtain the linearized variational equation:

xi + Saw ut + But = r(U).

Next we assume that this system can be transformed to

3 dn 8

ANU) sd + DAU) sa + Bg = F, (2.30)

i=1 t

where the A!’s are symmetric, and A°® is positive definite Such a simultaneous symmetrization is

possible for the gasdynamics equations In Appendix A we provide examples, including a newform we derived Equation (2.30) is also linear in g*, from which it follows that:

ỏ (4Ê” A°q 2) thà 4°" Aig S) + 4 (B+B*)¿

By integrating this equation and defining ||q®||3 = (4°,A4Ê) = fo q®*A°q® dx, we obtain an

esti-mate for g®:

+ (4°,A24°) + 2(4°,F)

<&¡|l# lễ + FIR — fa Ana? ds, 231)

where 7 = (m, ,z) 1s the outward unit vector normal to T, A, = yi, Ain, and

Gy = ||A)+V:A—B— #!||+1 (2.32)

We see from (2.31) that the sign of §- g°"Ang® ds is important If boundary conditions are given

such that this term is non-negative, well-posedness follows immediately:

t

IA OIA < #*2lể(@lä + ff era 2.33)

The new constant œ;, has absorbed the constant relating the norms || : ||4 and || : ||2 If we view

U as the computed solution, g* as the error and F as the residual, then the above inequality alsogives us an estimate of the computational error If the boundary term is not positive, then from thisanalysis we cannot make any conclusions on the well-posedness of the problem or convergence

of the approximation

Serrin used similar norm estimates to establish uniqueness for compressible flows [OS78,

Ser59}

2.4 Well Posedness Through Entropy Functions

By using entropy functions, we can generalize the concept of well-posedness to the nonlinearregime without linearizing the problems The entropy framework allows us to formulate differ-ence schemes that inherit the (generalized) well-posedness of the continuous problem Hence theschemes are stable in this new setting

2.4.1 Some Definitions

We first collect some definitions pertinent to all subsequent discussions For these and other

standard definitions, see [Ols95c, Tad97] and the references therein

Definition 2.4.1 A function f(u) is homogeneous of order B with respect to its argument u if

ƒ(Bu) = 0° f(w) Such a function satisfies the differential equation

fu = BF(u) (2.34)

Definition 2.4.2 The conservative system (2.23) is called symmetrizable if there exists a smooth

transformation represented by w= W (u) ©® u= U(w) and f'(u(w)) = f'(w), where the Jacobians U,, and fi, are symmetric, and Ủy is positive definite’.

Definition 2.4.3 The conservative system (2.23) has a skew-symmetric form if there exists asmooth transformation w = W(u) where W,, is positive definite, and there exist smooth functions

Y(w), G'(w) where Yy and GÌ, are symmetric, such that

g ở 9

ỏ i ỏ i i ở i — fi

oa = By, (Ow) + Gua, đ(0) = ƒ(0) (2.36)

Definition 2.4.4 A scalar function n(u) is called an entropy function for (2.23) if its Hessian

Nuu is positive definite and symmetrizes the Jacobians, fi(u) in the following sense:

Tuu(1)f2(w) = (Ful) Muu), i=1, d.

Let G = (q', ,q7) We say that (n,q) is an entropy pair for the conservation law if

—n(u) + © ử (u) = 0 (2.37)

As a consequence, the q'’s satisfy the compatibility relations

uu)’ fi(u) = gi(u), i=1, ,d (2.38)

4Note that W, = 051, and thus W, is also positive definite

Definition 2.4.5 A function u is called an entropy solution to (2.23) if, for all convex entropy

functions T\(w) and corresponding entropy flux functions q!’s,

2n) + Seg) <0 (2.39)

ot = Ox; ~ ,

Theorems by Godunov and Mock show the equivalence of symmetrizability and the existence

of entropy pairs (n,đ) [God61, Moc80] For a symmetrizable system, Tadmor showed that if the

entropy function 1\ is homogeneous, then the system has a skew-symmetric form [Tad84] Olssonthen proved that a symmetrizable system can always be cast in skew-symmetric form, and the

functions Y(w) and G'(w) which are needed in equations (2.35 — 2.36) are given by [Ols95c]:

Y(w) = / _(êu)d9, GÍ(w) = / Ộ #(@w)49 (2.40)

By applying this equivalence and using a suitably chosen transformation or entropy tion, Olsson showed how to obtain energy estimates for the solution u of (2.23) which echo theestimates in the linear case definition of well-posedness (see Definition 2.1.1)

func-2.4.2 Energy Estimates for the Analytic Solution

What follows is a generalization of Olsson’s analysis to include multiple dimensions and fluxfunctions that do not necessarily have the same order of homogeneity,

If we assume that (2.23) is symmetrizable, then a skew-symmetric form exists Further, if thetransformation in Definition 2.4.2 is homogeneous, then the equations in (2.40) reduce to

Y0)=g 100) GO) Ba fiw), (2.41) = aS

where B, and Bi, are the orders of homogeneity of U and f*, respectively Consequently, the

skew-symmetric forms of the derivatives also simplify to

the original system (2.23) as follows:

Finally, by the homogeneity of ƒ,

iwi = —(8u+ 1) wh (SS fr (2.45)

dt U , Si {Be +1 “le

As before, 7 = (nj, ,nq) is the outward unit normal to T

Before we can obtain a “true” energy estimate, we need to cast equation (2.45) in terms of the

data By the symmetry of each ƒƒ, there exists an orthogonal Q such that

where A; = diag(Ay, Am,), and A; < 0 for 1 < j < my These are the eigenvalues corresponding

to the the ingoing characteristic variables” Similarly, Ag contains the eigenvalues corresponding

to outgoing characteristics Equation (2.45) then becomes

where we have utilized the relationship wi AoVo >0 Since Ủ„ and A; both depend on the

solution, inequality (2.52) is not an energy estimate in the standard sense However, recall that

both are positive definite Therefore, all the terms in the estimate are positive functions of w

Below, we examine more closely what this generalized energy estimate means, and in particularfor fluid flow equations

5They are negative, since 7 is defined as an outward normal

ẾIn the discrete case discussion, we make clear the significance of this assumption

2.4.3 Entropy Well-posedness

The transformation needed to write a symmetrizable system into a skew-symmetric form is

re-lated to the entropy function by the equation w(u) = n,,(u) [Moc80, Har83] Once we choose atransformation or entropy function’, a straightforward calculation yields

w' O,,w = Cin(0), uˆWuu =n (u), (2.53)

where the constants C; and C2 come from the parameters defining n Thus, the generalized energy

l|w(-,f) l§ is the total or global entropy of the system Further, the flux functions q! associated with

1 satisfy wf /2w/(B + 1) = đ'(u) Hence the generalized energy equation (2.45), or equivalently

(2.48), simplifies to

d TaTnG)4 = —ÿñ-ñás (2.54)

which is simply a statement of global entropy conservation

The symmetrizability requirement is not overly severe Many problems from applications isfy this property including the equations for gasdynamics and magneto-hydrodynamics [Har83].Tadmor also showed how to handle the inhomogeneous flux functions of the compressible shallowwater equations to cast them into a more general skew-symmetric form [Tad84]

sat-We can obtain the equation for entropy conservation given by (2.54) directly from Definition

2.4.4 without needing the skew-symmetric form (2.44) and energy method analysis However,

the latter process carries over to the discrete case, and therefore allows us to formulate difference

schemes for initial-boundary value problems which then inherit this property of entropy

conser-vation We note that the new system (2.44) is no longer in conservative form; likewise its discreteversion In subsequent chapters, we discuss the effects of this

If we consider the existence of estimate (2.52) as entropy well-posedness for the continuousproblem (2.23 — 2.26), then its discrete version is entropy stability for numerical approximations.With this framework in mind, we formalize entropy well-posedness with the following definition:

Definition 2.4.6 The initial-boundary value problem defined by (2.23 — 2.26) is entropy posed if there exists a unique solution which satisfies the estimate given by (2.52)

well-In the discrete case, we need a stronger version of the above definition when we extend theframework for nonlinear stability and delve into error estimation This motivates the following

definition:

7This is assumed to be homogeneous with respect to the conservative variables

Definition 2.4.7 The initial-boundary value problem defined by (2.23 — 2.26) is entropy stronglywell-posed in the time interval |D,T) if there exists a unique solution which satisfies the estimategiven by

Iw, Dz < FOWT) (2.55)

where F is a function of the the initial data 6 and boundary data V

This says that the total entropy at time T can be bounded in terms of initial and boundary dataonly The class of problems for which we can show strong well-posedness in the entropy is morelimited Periodic problems and applications where characteristics’ behavior at boundaries areknown in a finite time interval do fall into this class

In their derivations of the skew-symmetric form from the entropy function, Tadmor [Tad84]

and Olsson [Ols95c] exploited the positivity of Ở„ (see Definition 2.4.2) This means that in order

to obtain the generalized energy estimate (2.52), Uy must be positive definite Further, because

of equation (2.53), the condition U,, > 0 automatically imposes a strict positivity (or negativity,depending on the sign of the constant C¡) on the entropy function: Vu #0, Cin(u) > 0 We

therefore have:

Lemma 2.4.1 JƒT\(u) is an entropy function associated with the conservative system (2.23), then

Vụ €R”, either n(u) > 0 or n(u) <0 Further, n(u) =0 ® u=0,

Definition 2.4.6 and Lemma 2.4.1 have implications for the choice of the entropy function nthat symmetrizes the system We analyze them below for the Euler gasdynamics equations

Case: The Euler equations for gasdynamics

For the Euler equations, Harten suggested two candidates for n(u), both of which are based on

the physical entropy, 5, of the system [Har83] We examine here their feasibility Following the

notation of Appendix A, let p,p,¥,£ denote the density, pressure, velocity and energy variables,respectively The entropy is then given by 5 = c, log(pp~”), where cy is the specific heat at con-

stant volume, and ¥ is the adiabatic exponent [LR57, Fas62] In keeping with Harten’s notation,let S = §/cy.

The first entropy function is given by: —pS Its corresponding flux is equal to —pS¥ This n

is not homogeneous with respect to the conservative variables, and consequently, does not lead to

a split-form (2.44)