Finite volume relaxation schemes for multidimensional conservation laws

Katsaounis, Theodoros; Makridakis, Charalambos

doi:10.1090/S0025-5718-00-01188-1

Finite volume relaxation schemes for multidimensional conservation laws

Authors: Theodoros Katsaounis and Charalambos Makridakis
Journal: Math. Comp. 70 (2001), 533-553
MSC (2000): Primary 65M12, 65M15; Secondary 65L06
DOI: https://doi.org/10.1090/S0025-5718-00-01188-1
Published electronically: March 3, 2000
MathSciNet review: 1681104
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Abstract: We consider finite volume relaxation schemes for multidimensional scalar conservation laws. These schemes are constructed by appropriate discretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a rate of $h^{1/4}$ in $L^{\infty }([0, T] , L^{1} _\mathrm{loc}({\mathbb{R}} ^{d} ))\,$ .

[BP] F. Bouchut and B. Perthame, Kružkov’s estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2847–2870 (English, with English and French summaries). MR 1475677, https://doi.org/10.1090/S0002-9947-98-02204-1
[CLL] Gui Qiang Chen, C. David Levermore, and Tai-Ping Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), no. 6, 787–830. MR 1280989, https://doi.org/10.1002/cpa.3160470602
[C] Bernardo Cockburn, On the continuity in 𝐵𝑉(Ω) of the 𝐿²-projection into finite element spaces, Math. Comp. 57 (1991), no. 196, 551–561. MR 1094943, https://doi.org/10.1090/S0025-5718-1991-1094943-9
[CCL] Bernardo Cockburn, Frédéric Coquel, and Philippe LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), no. 207, 77–103. MR 1240657, https://doi.org/10.1090/S0025-5718-1994-1240657-4
[CGa] B. Cockburn and H. Gau, A posteriori error estimates for general numerical methods for scalar conservation laws, Mat. Apl. Comput. 14 (1995), 37-47. CMP 95:15
[CG1] Bernardo Cockburn and Pierre-Alain Gremaud, Error estimates for finite element methods for scalar conservation laws, SIAM J. Numer. Anal. 33 (1996), no. 2, 522–554. MR 1388487, https://doi.org/10.1137/0733028
[CG2] Bernardo Cockburn and Pierre-Alain Gremaud, A priori error estimates for numerical methods for scalar conservation laws. I. The general approach, Math. Comp. 65 (1996), no. 214, 533–573. MR 1333308, https://doi.org/10.1090/S0025-5718-96-00701-6
[CG3] Bernardo Cockburn and Pierre-Alain Gremaud, A priori error estimates for numerical methods for scalar conservation laws. II. Flux-splitting monotone schemes on irregular Cartesian grids, Math. Comp. 66 (1997), no. 218, 547–572. MR 1408372, https://doi.org/10.1090/S0025-5718-97-00838-7
[CGY] Bernardo Cockburn, Pierre-Alain Gremaud, and Jimmy Xiangrong Yang, A priori error estimates for numerical methods for scalar conservation laws. III. Multidimensional flux-splitting monotone schemes on non-Cartesian grids, SIAM J. Numer. Anal. 35 (1998), no. 5, 1775–1803. MR 1640021, https://doi.org/10.1137/S0036142997316165
[CM] Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, https://doi.org/10.1090/S0025-5718-1980-0551288-3
[EG] R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by the finite volume schemes, IMA J. Numer. Anal. 18 (1998), 563-594. CMP 99:11
[GM] L. Gosse and Ch. Makridakis, A posteriori error estimates for numerical approximations to scalar conservation laws: Schemes satisfying strong and weak entropy inequalities. FORTH-IACM Technical Report 98.4 (1998).
[JX] Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276. MR 1322811, https://doi.org/10.1002/cpa.3160480303
[KM] Th. Katsaounis and Ch. Makridakis, Finite volume relaxation schemes for multidimensional conservation laws,, Preprint 97-12, Dept of Math., Univ. of Crete (1997).
[KZ] Th. Katsaounis and G. Zouraris, Numerical evaluation of relaxation schemes for multidimensional conservation laws, (To appear).
[KKM] M.A. Katsoulakis, G. Kossioris and Ch. Makridakis, Convergence and error estimates of relaxation schemes for multidimensional conservation laws, Comm. Partial Differential Equations 24 (1999), 395-424. CMP 99:11
[KT1] Markos A. Katsoulakis and Athanasios E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations 22 (1997), no. 1-2, 195–233. MR 1434144, https://doi.org/10.1080/03605309708821261
[KR] Dietmar Kröner and Mirko Rokyta, Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions, SIAM J. Numer. Anal. 31 (1994), no. 2, 324–343. MR 1276703, https://doi.org/10.1137/0731017
[Kr] S.N. Kruzhkov, First order quasilinear equations with several independent variables, Math. USSR Sbornik 10 (1970), 217-243.
[Kz] N.N. Kuznetzov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comp. Math. and Math. Phys. 16 (1976), 105-119.
[N1] Denis Serre, Problèmes de Riemann singuliers, Appl. Anal. 35 (1990), no. 1-4, 175–185 (French, with English summary). MR 1387407, https://doi.org/10.1080/00036819008839915
[N2] Roberto Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Differential Equations 148 (1998), no. 2, 292–317. MR 1643175, https://doi.org/10.1006/jdeq.1998.3460
[R] Christian Rohde, Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D, Numer. Math. 81 (1998), no. 1, 85–123. MR 1657726, https://doi.org/10.1007/s002110050385
[S] M. Kh. Dorri, Approximate solution of systems of hyperbolic equations, based on interpolation of a residue in Hermite’s sense, Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), no. 6, 1410–1420, 1533 (Russian). MR 683598
[ScTW] Hans Joachim Schroll, Aslak Tveito, and Ragnar Winther, An 𝐿¹-error bound for a semi-implicit difference scheme applied to a stiff system of conservation laws, SIAM J. Numer. Anal. 34 (1997), no. 3, 1152–1166. MR 1451118, https://doi.org/10.1137/S0036142994268855
[ShTW] Wen Shen, Aslak Tveito, and Ragnar Winther, A system of conservation laws including a stiff relaxation term; the 2D case, BIT 36 (1996), no. 4, 786–813. MR 1420276, https://doi.org/10.1007/BF01733792
[V] J.-P. Vila, Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes, RAIRO Modél. Math. Anal. Numér. 28 (1994), no. 3, 267–295 (English, with English and French summaries). MR 1275345, https://doi.org/10.1051/m2an/1994280302671