On numerical entropy inequalities for a class of relaxed schemes

Tang, Huazhong; Tang, Tao; Wang, Jinghua

doi:10.1090/qam/1828460

Authors: Huazhong Tang, Tao Tang and Jinghua Wang
Journal: Quart. Appl. Math. 59 (2001), 391-399
MSC: Primary 65M06; Secondary 35B25, 35L65
DOI: https://doi.org/10.1090/qam/1828460
MathSciNet review: MR1828460
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In [4], Jin and Xin developed a class of first- and second-order relaxing schemes for nonlinear conservation laws. They also obtained the relaxed schemes for conservation laws by using a Hilbert expansion for the relaxing schemes. The relaxed schemes were proved to be total variational diminishing (TVD) in the zero relaxation limit for scalar equations. In this paper, by properly choosing the numerical entropy flux, we show that the relaxed schemes also satisfy the entropy inequalities. As a consequence, the ${L^{1}}$ convergence rate of $O\left ( \sqrt {\Delta t} \right )$ for the relaxed schemes can be established.

Denise Aregba-Driollet and Roberto Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996), no. 1-2, 163–193. MR 1625520, DOI https://doi.org/10.1080/00036819608840453
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, DOI https://doi.org/10.1090/S0025-5718-96-00701-6
Shi Jin, A convex entropy for a hyperbolic system with relaxation, J. Differential Equations 127 (1996), no. 1, 97–109. MR 1387259, DOI https://doi.org/10.1006/jdeq.1996.0063
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, DOI https://doi.org/10.1002/cpa.3160480303
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, DOI https://doi.org/10.1080/03605309708821261
A. Kurganov and E. Tadmor, Stiff systems of hyperbolic conservation laws: convergence and error estimates, SIAM J. Math. Anal. 28 (1997), no. 6, 1446–1456. MR 1474223, DOI https://doi.org/10.1137/S0036141096301488
Hailiang Liu and Gerald Warnecke, Convergence rates for relaxation schemes approximating conservation laws, SIAM J. Numer. Anal. 37 (2000), no. 4, 1316–1337. MR 1756752, DOI https://doi.org/10.1137/S0036142998346882
Roberto Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), no. 8, 795–823. MR 1391756, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199608%2949%3A8%3C795%3A%3AAID-CPA2%3E3.0.CO%3B2-3
Roberto Natalini, Recent results on hyperbolic relaxation problems, Analysis of systems of conservation laws (Aachen, 1997) Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 99, Chapman & Hall/CRC, Boca Raton, FL, 1999, pp. 128–198. MR 1679940
Haim Nessyahu and Eitan Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), no. 2, 408–463. MR 1047564, DOI https://doi.org/10.1016/0021-9991%2890%2990260-8
Hua-zhong Tang and Hua-mo Wu, On a cell entropy inequality of the relaxing schemes for scalar conservation laws, J. Comput. Math. 18 (2000), no. 1, 69–74. MR 1741174
Eitan Tadmor and Tao Tang, Pointwise error estimates for relaxation approximations to conservation laws, SIAM J. Math. Anal. 32 (2000), no. 4, 870–886. MR 1814742, DOI https://doi.org/10.1137/S0036141098349492
Zhen-Huan Teng, First-order $L^1$-convergence for relaxation approximations to conservation laws, Comm. Pure Appl. Math. 51 (1998), no. 8, 857–895. MR 1620220, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199808%2951%3A8%3C857%3A%3AAID-CPA1%3E3.3.CO%3B2-M
Jinghua Wang and Gerald Warnecke, Convergence of relaxing schemes for conservation laws, Advances in nonlinear partial differential equations and related areas (Beijing, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 300–325. MR 1690837

W. A. Yong, Numerical analysis of relaxation schemes for scalar conservation laws, Technical Report 95-30 (SFB 359), IWR, University of Heidelberg, 1995.