Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

On numerical entropy inequalities for a class of relaxed schemes


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
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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.


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  • [1] D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61, 163-193 (1996) MR 1625520
  • [2] B. Cockburn and P.-A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach, Math. Comp. 65, 533-573 (1996) MR 1333308
  • [3] S. Jin, A convex entropy for a hyperbolic system with relaxation, J. Differential Equations 127, 95-107 (1996) MR 1387259
  • [4] S. Jin and Z.-P. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48, 235-281 (1995) MR 1322811
  • [5] M. A. Katsoulakis and A. E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations 22, 195-233 (1997) MR 1434144
  • [6] A. Kurganov and E. Tadmor, Stiff systems of hyperbolic conservation laws, convergence and error estimates, SIAM J. Math. Anal. 28, 1446-1456 (1997) MR 1474223
  • [7] H. L. Liu and G. Warnecke, Convergence rates for relaxation schemes approximating conservation laws, SIAM J. Numer. Anal. 37, 1316-1337 (2000) MR 1756752
  • [8] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49, 795-824 (1996) MR 1391756
  • [9] R. Natalini, Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws (Aachen, 1997), 128-198, Chapman and Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 99, Chapman and Hall/CRC, Boca Raton, FL, 1999 MR 1679940
  • [10] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87, 408-463 (1990) MR 1047564
  • [11] H.-Z. Tang and H.-M. Wu, On a cell entropy inequality for the relaxing schemes of scalar conservation laws, J. Comput. Math. 18, 69-74 (2000) MR 1741174
  • [12] E. Tadmor and T. Tang, Pointwise error estimates for relaxation approximations to conservation laws, SIAM J. Appl. Math. (to appear) MR 1814742
  • [13] Z.-H. Teng, First-order $ L^{1}$ -convergence for relaxation approximations to conservation laws, Comm. Pure Appl. Math. 51, 857-895 (1998) MR 1620220
  • [14] J. Wang and G. Warnecke, Convergence of relaxing schemes for conservation laws, Advances in Nonlinear Partial Differential Equations and Related Areas, G.-Q. Chen, Y. Li, X. Zhu, and D. Cao (eds.), World Scientific Publishing, River Edge, NJ, 1998, pp. 300-325 MR 1690837
  • [15] W. A. Yong, Numerical analysis of relaxation schemes for scalar conservation laws, Technical Report 95-30 (SFB 359), IWR, University of Heidelberg, 1995.

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DOI: https://doi.org/10.1090/qam/1828460
Article copyright: © Copyright 2001 American Mathematical Society

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