Luận án tiến sĩ: Ổn định và ước lượng sai số dùng hàm entropy Stanford

STABILITY AND ERROR ESTIMATION USING ENTROPY FUNCTIONS A DISSERTATION SUBMITTED TO THE PROGRAM IN SCIENTIFIC COMPUTING AND COMPUTATIONAL MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Melissa D. Aczon December 2006 UMI Number: 3242513 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted.

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ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 © Copyright by Melissa D. Aczon 2007 All Rights Reserved ii 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. a“ 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 (Doron Levy) Cf Approved for the University Committee on Graduate Studies. 1H Abstract 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 obtain rigorous 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 splitting process 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 and the 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 each other.

Our results demonstrate the existence of an optimal value which minimizes errors. Both our theoretical analysis and computational examples show advantages of using the split schemes over their un-split counterparts. These benefits include greater accuracy and efficiency for the solution 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 of both the stability estimates and entropy errors provides insights into these behaviors. We also derived an error estimate for the entropy function of symmetrizable hyperbolic sys- tems. Like classical estimates for the computational error, the estimate for the entropy error reflects an accumulation of local truncation errors. We show that the computational and entropy errors converge in a similar fashion.

Therefore, the entropy error estimate can be used to monitor and control global accuracy. Since the entropy error estimate utilizes variables already computed by the discretization kernel, it does not add any substantial cost to the solution algorithm. We also demonstrate the feasibility of using the entropy error estimate as a monitor function for local grid adaption purposes. iv Acknowledgements Many people have supported, influenced and enriched me over the years.

It is my pleasure to acknowledge 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 seemingly inconsequential, and from afar. Then you challenged and pushed my understanding to levels beyond what I had imagined.

Words are inadequate to express how truly grateful I am to you. Cheers! e Joseph Oliger — You supported and grounded me for so many years. Thank you for sharing your time and pearls of wisdom. I feel privileged to have known and worked with you.

e Doron Levy and Robert Street — Thank you for all the time you spent answering my ques- tions and reading my thesis. e Pelle Olsson and Berti! Gustafsson — Thank you for answering my questions 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 and fue] 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 reminders that there is so much more to life than matrices and PDEs. e Clifford Stein — You are everything to me, from my technical administrator to my emotional rock.

There is no one whose patience, understanding, intelligence and integrity I admire more than I do yours. Above all, I treasure your love, for it continues to carry me through life. I wish to thank the National Science Foundation, whose grant (2DMA444) partially supported this work. Finally, I wish to dedicate this work to the memory of Stavros Busenberg, my undergraduate math mentor and advisor.

He introduced me to a new world filled with adventures I had never before imagined. He challenged and encouraged me. His joyful and passionate embrace of life — from its simple delights to its deep mathematical conundrums — has continued to inspire me over the years, especially when I have doubted my paths. vi Contents Abstract iv Acknowledgements Y 1 Introduction 1 1.1 Background and Motivation .2 Stability and Error Estimation.

Q Q HQ eee eee 2 1.1 Computational Efficiency: Grid Adaption.2 Reliability: Overall Robustness of Simulation.3 Grid Quality and Control. ee 3 143 Scope of Dissertation. Q HQ HH ko 4 1.4 Overview and Summary of Results. es 5 2 The Analytic Problem and Numerical Scheme 2.1 The Analytic Problem and Well-posednes.

HH gà gà kà k Ra 2. 1n nh h TH A4 ee ee 11 2.3 Hyperbolic Conservation Laws.4 Well Posedness Through Entropy Functions. eee ee es 18 24. Q Q ee ee ee 19 vii 2.2 Energy Estimates for the Analytic Solution.

Entropy Well-posedness. ee ee eee 23 2.5 Nonlinear Stability Through Entropy. eee ee ee 26 2.1 The Semi-discrete System .2 Generalized Stability Estimates .6 The Numerical Scheme.00 0c Q eee ee eee 32 2.1 Semi-discrete Form.2 Time-stepping Methods .3 Numerical Boundary Condilons.7 Euler Test Problems.1 Traveling Isentropic Vortex.2 Advection of Density Wave (Hump). HQ HQ HQ nạ n kg kh k kia 36 Effect of Splitting Parameter 39 3.2 Numerical Comparisons of Local Accuracy .3 Long Time Integration 2.4 Evolution of Computational Error.

Initial-Boundary Value Problem. ó1 Entropy Conservation and Errors 64 4.1 Continuous and Semi-discrete Entropy Estimates. 64 42 Sources of Entropy Errors 2. Convergence of the Global Entropy.1 Errors Incurred by not Splitting .4 Convergence of Other Errors.4 Comparison of Split and Unsplit Entropy Erors.

Q HQ Q HQ Quà kg VN va 4.5 Discontinuities and the Entropy Solution. HQ HQ và ee "` )''° “d4. Euler System: Contact Discontinuity .1 Linearized Error Estimates.2 Cockburn and Gau’s Estimate for Scalar Equations. Entropy Error Estimate for Systems .1 Convergence of Errors: Smooth Case.

Convergence of Errors: Non-Smooth .4 Global Error Estimation.5 On the Sharpness of the Entropy Error Estimate.1 Effectof Norm Used. 2 ee ee ee 119 5.6 Local Error Estimation ©.1 Local Comparisons of Errors.2 Comparison of Some Monitor Functions.1 Entropy Stability and Splitting .2 Error Estimation with Entropy Functions.0 eee eee 130 The Euler Equations of Gasdynamics A.l The Conservative Form.2 Linearized Equations and Simultaneous Symmetrization. 6 6 HH HH Quà v. Q Q Q Q HH go 1X B_ Difference Operators 139 B.1 Spatial Operators D and Corresponding & Norms CS — .- 139 C Local Truncation and Residual Errors 142 C.1 Richardson Extrapolation CS SS SS SS 144 References 146 List of Tables 2.1 2D Vortex, RK-TVD: r;, corresponding to Da,D¿,CFL=01.2 2D Vortex, RK-TVD: 7; corresponding to 2Da,D¿,CEL=035.3 1D, Heun’s: rg corresponding to D4, Dg, CTEFL=0.4 2D Vortex, Heun’s: r; corresponding to Da, Dạ, CFLEO.5 2D Vortex, Forward Euler: r¿ corresponding to Đa, Dg, CFL=0.1 Stopping Times: 1D hump, RK-TVD,Order=6,N=Z2l.2 Stopping Times: 2D Vortex, RK-TVD, Order=6,N=2l.3 Break Times 7,: RK-TVD, Vortex strengthe =6.1 ry(€4(Gj)) and ef: ID Hump, Dạ, N=40,T=0.2 ry(En (ô¿)) and ef’: 1D Hump, Ds, W=20,T=05.3 r¿(En (64) and ef’: 2D Vortex, Dạ, N= 40, T=0.4 r¿(En (ô;)) and ey: 2D Vortex, Dạ, N= 20, T=0Ú.5 Convergence Order: 1D Hump, RK-TVD, D2, T=0.6 Convergence Order: 2D vortex, Dạ, 7 =0.7 €,(&;) and ey: ID,pfƒ, Dạ, N=100,T=O1 .9 rg: Rarefaction; No = 20, Dp, CFL=0.10 r„: Shock; No = 20, Dạ, CFL=0.11 rg: 1D, RK-TVD-3, CFL=0.12 Rarefaction 1: Cr with U, = 0, Up =1, Ax =0.13 Rarefaction 2: Cr with U; = —1, Ur = 1, Ax=0.14 Shock 1: Cr with Up = 1, Up = —1, Dạ, CFL=05 2.15 Shock 1: Cr with Up =1, Ur =—1, Dạ,CFL=08.16 Shock location for each vụ: Up = 1, UR=0, Da.17 Shock 2: Cr with Ứy = 1, Ur =0, Dạ, CFL=0.

ee ee ee 101 4.18 Shock 2: Cr with U, = 1, Up =0, D4, CFL=08 2.19 Shock 3 Cr with U, = 3 Up = 1, Dạ, CFL=0.20 Shock 3 Cr with U, = 3 Ủa = 1, Dạ, CFL=05.21 Shock locations for each v„(7): Up = 3, Us=1,Da,T=04.22 Comparison of Cr: py = 10.1 Convergence Comparison of computational and entropy errors .2 Convergence Comparison of computational and entropy errors .3 Convergence Comparison of computational and entropy errors .4 Convergence Comparison of computational and entropy errors. eee 122 Xil List of Figures 3.1 c¡ and ca as functions of &@ 2. QC Q LH Q HH ng Q2 vkv v2 40 3.2 Behavior of |p*| as a function ofÔ_.3 Spectral Radius Behavior.4 Behavior of gt Am.5 ID: Ì|rs||~ and ||tụ ||~ : pậ, © = 2, various vo, Ö. and ||ty || 92, De, N = 40, low strengthe.

pạ, De, N= 40, highe values .17 lle(t;6) ||, v = (1,0), Order=4, N=40, CFL=0.19 llcứ: &) ||, v = (1,0), Order=6, N=40, CFL=0.24 ||e(&)|| 1D hump, T=30; Order=6, CFL =0.25 ||@(&) || 2D hump, Order=6, CFL= 0.26 lle(&)l| p¥, Order=4,CFL=0.27 lle(&)|| pi, Order=4, CFL =0.1 Erp — 1D, RK-TVD, Dạ, N=80,TH=1.2 e¡(ô; Ar) /A% - 1D, RK-TVD, De, N=80,T=10.3 En, - ID, RK-TVD, Dạ, N=80, T= 1Ú.4 Erp - RK-TVD, Do, N= 80, T=10.5 &4 (6; Ar) /fiỆ — 1D, RK-TVD, Dp N= 80, T=10.6 &n, - ID, RK-TVD, Dg, N= 80, T=10. eee ee eee 82 4.7 Erg — 1D, Heun, Dạ, N= 80, TH1.8 e(G; At) /ñỂ — ID, Heun, Dg, N=80, T=1.9 En, — 1D, Heun, Ds, N=80, T=1.10 Err — 1D, Heun, Dg, N=80, T=10.11 £5 (6; Ar) /ñỆ - 1D, Heun, Ds, W= 80, T=10.13 Erg — 2D Vortex, RK-TVD-3, De, N=40,7=10.14 €4 (6; At) /fiỆ - 2D Vortex, RK-TVD-3, Ds, N=40,7=10.15 en, - 2D Vortex, RK-TVD-3, Dg, N=40,T7=l0.16 Erp — 2D Vortex, RK-TVD-3, Dạ, N=40,T7=30.17 e¡ (ô; At) /ñỆ - 2D Vortex, RK-TVD-3, Ds, N=40,T7=30.18 En, - 2D Vortex, RK-TVD-3, Dp, N=40, T=30 2.19 Erg — 2D Vortex, Heun, Dg, N=40,T=10 ee ©.