A possible counterexample to well posedness of entropy solutions and to Godunov scheme convergence
Author:
Volker Elling
Journal:
Math. Comp. 75 (2006), 17211733
MSC (2000):
Primary 35L65, 35L67, 76L05, 76H05, 76N10
Published electronically:
June 19, 2006
MathSciNet review:
2240632
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A particular case of initial data for the twodimensional Euler equations is studied numerically. The results show that the Godunov method does not always converge to the physical solution, at least not on feasible grids. Moreover, they suggest that entropy solutions (in the weak entropy inequality sense) are not well posed.
 [BB01]
S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Tech. report, S.I.S.S.A., Trieste, Italy, 2001.
 [BCP00]
A. Bressan, G. Crasta, and B. Piccoli, Wellposedness of the Cauchy problem for systems of conservation laws, Memoirs AMS, no. 694, American Mathematical Society, July 2000.
 [BG99]
A. Bressan and P. Goatin, Oleinik type estimates and uniqueness for conservation laws, J. Diff. Eqs. 156 (1999), 2649.
 [BL97]
A. Bressan and P. LeFloch, Uniqueness of weak solutions to systems of conservation laws, Arch. Rat. Mech. Anal. 140 (1997), 301317.
 [CF48]
R. Courant and K.O. Friedrichs, Supersonic flow and shock waves, Interscience Publishers, 1948.
 [CL81]
J.G. Conlon and TaiPing Liu, Admissibility criteria for hyperbolic conservation laws, Indiana Univ. Math. J. 30 (1981), no. 5, 641652.
 [Daf00]
C. Dafermos, Hyperbolic conservation laws in continuum physics, Springer, 2000.
 [EL05]
V. Elling and TaiPing Liu, The ellipticity principle for selfsimilar potential flow, J. Hyper. Diff. Eqns. (2005), to appear, Preprint arxiv:math.AP0509332.
 [Ell00]
V. Elling, Numerical simulation of gas flow in moving domains, Diploma Thesis, RWTH Aachen (Germany), 2000.
 [Ell03]
, A LaxWendroff type theorem for unstructured quasiuniform grids, 2003, arxiv:math.NA/0509331.
 [Ell05]
, A LaxWendroff type theorem for unstructured grids, Ph.D. Dissertation, Stanford University, 2005.
 [Eva98]
L.C. Evans, Partial differential equations, American Mathematical Society, 1998.
 [Gli65]
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697715.
 [God59]
S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), 271290.
 [GR96]
E. Godlewski and P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Springer, 1996.
 [HHL76]
A. Harten, J.M. Hyman, and P.D. Lax, On finitedifference approximation and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297321.
 [Hop67]
E. Hopf, On the right weak solution of the Cauchy problem for quasilinear equations of first order, J. Math. Mech. 17 (1967), 483487.
 [Kru70]
S.N. Kruzkov, First order quasilinear equations in several independent variables, Mat. Sb. 81 (1970), no. 2, 285355, transl. in Math. USSR Sb. 10 (1970) no. 2, 217243.
 [KRW96]
D. Kröner, M. Rokyta, and M. Wierse, A LaxWendroff type theorem for upwind finite volume schemes in 2D, EastWest J. Numer. Math. 4 (1996), 279292.
 [Lax71]
P.D. Lax, Shock waves and entropy, Contributions to Nonlinear Functional Analysis (E.A. Zarantonello, ed.), Academic Press, 1971, pp. 603634.
 [Liu74]
TaiPing Liu, The Riemann problem for general conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89112.
 [Liu75]
, The Riemann problem for general systems of conservation laws, J. Diff. Eqs. 18 (1975), 218234.
 [Liu81]
, Admissible solutions of hyperbolic conservation laws, Memoirs AMS, no. 240, American Mathematical Society, 1981.
 [LL98]
P.D. Lax and XuDong Liu, Solution of twodimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19 (1998), no. 2, 319340.
 [LW60]
P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217237.
 [LY99]
TaiPing Liu and Tong Yang, Wellposedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (1999), 15531586.
 [LZY98]
Jiequan Li, Tong Zhang, and Shuli Yang, The twodimensional Riemann problem in gas dynamics, Addison Wesley Longman, 1998.
 [MO79]
A. Majda and S. Osher, Numerical viscosity and the entropy condition, Comm. Pure Appl. Math. 32 (1979), 797838.
 [OC84]
S. Osher and S. Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal. 21 (1984), no. 5, 955984.
 [OS82]
S. Osher and F. Solomon, Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comp. 38 (1982), 339373.
 [OT88]
S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), no. 181, 1951.
 [Ser99]
D. Serre, Systems of conservation laws, vol. 1, Cambridge University Press, 1999.
 [SO89]
C. W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shockcapturing schemes, II, J. Comp. Phys. 83 (1989), 3278.
 [Tad84]
E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984), no. 168, 369381.
 [Tad87]
, The numerical viscosity of entropy stable schemes for systems of conservation laws, I, Math. Comp. 49 (1987), no. 179, 91103.
 [BB01]
 S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Tech. report, S.I.S.S.A., Trieste, Italy, 2001.
 [BCP00]
 A. Bressan, G. Crasta, and B. Piccoli, Wellposedness of the Cauchy problem for systems of conservation laws, Memoirs AMS, no. 694, American Mathematical Society, July 2000.
 [BG99]
 A. Bressan and P. Goatin, Oleinik type estimates and uniqueness for conservation laws, J. Diff. Eqs. 156 (1999), 2649.
 [BL97]
 A. Bressan and P. LeFloch, Uniqueness of weak solutions to systems of conservation laws, Arch. Rat. Mech. Anal. 140 (1997), 301317.
 [CF48]
 R. Courant and K.O. Friedrichs, Supersonic flow and shock waves, Interscience Publishers, 1948.
 [CL81]
 J.G. Conlon and TaiPing Liu, Admissibility criteria for hyperbolic conservation laws, Indiana Univ. Math. J. 30 (1981), no. 5, 641652.
 [Daf00]
 C. Dafermos, Hyperbolic conservation laws in continuum physics, Springer, 2000.
 [EL05]
 V. Elling and TaiPing Liu, The ellipticity principle for selfsimilar potential flow, J. Hyper. Diff. Eqns. (2005), to appear, Preprint arxiv:math.AP0509332.
 [Ell00]
 V. Elling, Numerical simulation of gas flow in moving domains, Diploma Thesis, RWTH Aachen (Germany), 2000.
 [Ell03]
 , A LaxWendroff type theorem for unstructured quasiuniform grids, 2003, arxiv:math.NA/0509331.
 [Ell05]
 , A LaxWendroff type theorem for unstructured grids, Ph.D. Dissertation, Stanford University, 2005.
 [Eva98]
 L.C. Evans, Partial differential equations, American Mathematical Society, 1998.
 [Gli65]
 J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697715.
 [God59]
 S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), 271290.
 [GR96]
 E. Godlewski and P.A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Springer, 1996.
 [HHL76]
 A. Harten, J.M. Hyman, and P.D. Lax, On finitedifference approximation and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297321.
 [Hop67]
 E. Hopf, On the right weak solution of the Cauchy problem for quasilinear equations of first order, J. Math. Mech. 17 (1967), 483487.
 [Kru70]
 S.N. Kruzkov, First order quasilinear equations in several independent variables, Mat. Sb. 81 (1970), no. 2, 285355, transl. in Math. USSR Sb. 10 (1970) no. 2, 217243.
 [KRW96]
 D. Kröner, M. Rokyta, and M. Wierse, A LaxWendroff type theorem for upwind finite volume schemes in 2D, EastWest J. Numer. Math. 4 (1996), 279292.
 [Lax71]
 P.D. Lax, Shock waves and entropy, Contributions to Nonlinear Functional Analysis (E.A. Zarantonello, ed.), Academic Press, 1971, pp. 603634.
 [Liu74]
 TaiPing Liu, The Riemann problem for general conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89112.
 [Liu75]
 , The Riemann problem for general systems of conservation laws, J. Diff. Eqs. 18 (1975), 218234.
 [Liu81]
 , Admissible solutions of hyperbolic conservation laws, Memoirs AMS, no. 240, American Mathematical Society, 1981.
 [LL98]
 P.D. Lax and XuDong Liu, Solution of twodimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19 (1998), no. 2, 319340.
 [LW60]
 P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217237.
 [LY99]
 TaiPing Liu and Tong Yang, Wellposedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (1999), 15531586.
 [LZY98]
 Jiequan Li, Tong Zhang, and Shuli Yang, The twodimensional Riemann problem in gas dynamics, Addison Wesley Longman, 1998.
 [MO79]
 A. Majda and S. Osher, Numerical viscosity and the entropy condition, Comm. Pure Appl. Math. 32 (1979), 797838.
 [OC84]
 S. Osher and S. Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal. 21 (1984), no. 5, 955984.
 [OS82]
 S. Osher and F. Solomon, Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comp. 38 (1982), 339373.
 [OT88]
 S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), no. 181, 1951.
 [Ser99]
 D. Serre, Systems of conservation laws, vol. 1, Cambridge University Press, 1999.
 [SO89]
 C. W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shockcapturing schemes, II, J. Comp. Phys. 83 (1989), 3278.
 [Tad84]
 E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984), no. 168, 369381.
 [Tad87]
 , The numerical viscosity of entropy stable schemes for systems of conservation laws, I, Math. Comp. 49 (1987), no. 179, 91103.
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Additional Information
Volker Elling
Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
Email:
velling@stanfordalumni.org
DOI:
http://dx.doi.org/10.1090/S0025571806018631
PII:
S 00255718(06)018631
Keywords:
Conservation law,
well posedness,
entropy solution,
Riemann problem,
shock,
contact discontinuity,
compressible Euler equations,
entropy/entropy flux pair
Received by editor(s):
November 7, 2004
Received by editor(s) in revised form:
May 5, 2005
Published electronically:
June 19, 2006
Additional Notes:
This material is based upon work supported by an SAP/Stanford Graduate Fellowship and by the National Science Foundation under Grant no. DMS 0104019. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
