Highorder schemes and entropy condition for nonlinear hyperbolic systems of conservation laws
Author:
J.P. Vila
Journal:
Math. Comp. 50 (1988), 5373
MSC:
Primary 65M10; Secondary 35L65
MathSciNet review:
917818
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Abstract: A systematic procedure for constructing explicit, quasi secondorder approximations to strictly hyperbolic systems of conservation laws is presented. These new schemes are obtained by correcting firstorder schemes. We prove that limit solutions satisfy the entropy inequality. In the scalar case, we prove convergence to the unique entropysatisfying solution if the initial scheme is Total Variation Decreasing (i.e., TVD) and consistent with the entropy condition. Finally, we slightly modify Harten's highorder schemes such that they obey the previous conditions and thus converge towards the "entropy" solution.
 [1]
D. L. Book, J. P. Boris & K. Hain, "Flux corrected transport. II," J. Comput. Phys., v. 18, 1975, pp. 248283.
 [2]
Ronald
J. DiPerna, Uniqueness of solutions to hyperbolic conservation
laws, Indiana Univ. Math. J. 28 (1979), no. 1,
137–188. MR
523630 (80i:35119), http://dx.doi.org/10.1512/iumj.1979.28.28011
 [3]
Ami
Harten, High resolution schemes for hyperbolic conservation
laws, J. Comput. Phys. 49 (1983), no. 3,
357–393. MR
701178 (84g:65115), http://dx.doi.org/10.1016/00219991(83)901365
 [4]
Amiram
Harten and Peter
D. Lax, A random choice finite difference scheme for hyperbolic
conservation laws, SIAM J. Numer. Anal. 18 (1981),
no. 2, 289–315. MR 612144
(83b:65090), http://dx.doi.org/10.1137/0718021
 [5]
A. Harten, P. D. Lax & B. VanLeer, Upstream Differencing and Godunov Type Schemes for Hyperbolic Conservation Laws, ICASE, 825.
 [6]
A.
Harten, J.
M. Hyman, and P.
D. Lax, On finitedifference approximations and entropy conditions
for shocks, Comm. Pure Appl. Math. 29 (1976),
no. 3, 297–322. With an appendix by B. Keyfitz. MR 0413526
(54 #1640)
 [7]
Peter
Lax, Shock waves and entropy, Contributions to nonlinear
functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin,
Madison, Wis., 1971) Academic Press, New York, 1971,
pp. 603–634. MR 0393870
(52 #14677)
 [8]
A.
Y. le Roux, Numerical stability for some equations
of gas dynamics, Math. Comp.
37 (1981), no. 156, 307–320. MR 628697
(82m:76044), http://dx.doi.org/10.1090/S00255718198106286978
 [9]
A.
Y. LeRoux and P.
Quesseveur, Convergence of an antidiffusion LagrangeEuler scheme
for quasilinear equations, SIAM J. Numer. Anal. 21
(1984), no. 5, 985–994. MR 760627
(85m:65092), http://dx.doi.org/10.1137/0721061
 [10]
Andrew
Majda and Stanley
Osher, A systematic approach for correcting nonlinear
instabilities. The LaxWendroff scheme for scalar conservation laws,
Numer. Math. 30 (1978), no. 4, 429–452. MR 502526
(80g:65101), http://dx.doi.org/10.1007/BF01398510
 [11]
S. Osher & S. Chakravarthy, High Resolution Schemes and the Entropy Condition, ICASE 8349.
 [12]
E. Tadmor, Numerical Viscosity and the Entropy Condition for Conservative Difference Schemes, ICASE 8320.
 [13]
Eitan
Tadmor, The largetime behavior of the scalar,
genuinely nonlinear LaxFriedrichs scheme, Math. Comp. 43 (1984), no. 168, 353–368. MR 758188
(86g:65162), http://dx.doi.org/10.1090/S00255718198407581888
 [14]
J.P.
Vila, Simplified Godunov schemes for 2×2 systems of
conservation laws, SIAM J. Numer. Anal. 23 (1986),
no. 6, 1173–1192. MR 865949
(88d:65131), http://dx.doi.org/10.1137/0723079
 [1]
 D. L. Book, J. P. Boris & K. Hain, "Flux corrected transport. II," J. Comput. Phys., v. 18, 1975, pp. 248283.
 [2]
 R. J. DiPerna, "Uniqueness of solutions to hyperbolic conservation laws," Indiana Univ. Math. J., v. 28, 1979, pp. 137187. MR 523630 (80i:35119)
 [3]
 A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357393. MR 701178 (84g:65115)
 [4]
 A. Harten & P. D. Lax, "A random choice finite difference scheme for hyperbolic conservation laws," SIAM J. Numer. Anal., v. 18, 1981, pp. 289315. MR 612144 (83b:65090)
 [5]
 A. Harten, P. D. Lax & B. VanLeer, Upstream Differencing and Godunov Type Schemes for Hyperbolic Conservation Laws, ICASE, 825.
 [6]
 A. Harten, J. M. Hyman & P. D. Lax, "On finite difference approximations and entropy condition for shocks," Comm. Pure. Appl. Math., v. 29, 1976, pp. 297322. MR 0413526 (54:1640)
 [7]
 P. D. Lax, "Shock waves and entropy," in Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.), Academic Press, New York, 1971, pp. 603634. MR 0393870 (52:14677)
 [8]
 A. Y. Le Roux, "Numerical stability for some equations of gas dynamics," Math. Comp., v. 37, 1981, pp. 307320. MR 628697 (82m:76044)
 [9]
 A. Y. Le Roux & P. Quesseveur, "Convergence of an antidiffusion LagrangeEuler scheme for quasilinear equations," SIAM J. Numer. Anal., v. 21, 1984, pp. 985994. MR 760627 (85m:65092)
 [10]
 A. Majda & S. Osher, "A systematic approach for correcting nonlinear instabilities," Numer. Math., v. 30, 1978, pp. 429452. MR 502526 (80g:65101)
 [11]
 S. Osher & S. Chakravarthy, High Resolution Schemes and the Entropy Condition, ICASE 8349.
 [12]
 E. Tadmor, Numerical Viscosity and the Entropy Condition for Conservative Difference Schemes, ICASE 8320.
 [13]
 E. Tadmor, "The largetime behavior of the scalar, genuinely nonlinear LaxFriedrichs scheme," Math. Comp., v. 43, 1984, pp. 353368. MR 758188 (86g:65162)
 [14]
 J. P. Vila, "Simplified Godunov schemes for systems of conservation laws," SIAM J. Numer. Anal., v. 23, 1986, pp. 11731192. MR 865949 (88d:65131)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809178181
PII:
S 00255718(1988)09178181
Article copyright:
© Copyright 1988
American Mathematical Society
