On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms
HTML articles powered by AMS MathViewer
- by Abdallah Chalabi PDF
- Math. Comp. 66 (1997), 527-545 Request permission
Abstract:
We deal in this study with the convergence of a class of numerical schemes for scalar conservation laws including stiff source terms. We suppose that the source term is dissipative but it is not necessarily a Lipschitzian function. The convergence of the approximate solution towards the entropy solution is established for first and second order accurate MUSCL and for splitting semi-implicit methods.References
- F. Benkhaldoun and A. Chalabi, Characteristic based scheme for hyperbolic conservation laws with source terms, Submitted.
- A. C. Berkenbosch, E. F. Kaasschieter, and J. H. M. Ten Thije Boonkkamp, The numerical wave speed for one-dimensional scalar conservation laws with source terms. Preprint, Dept. of Math. and Comp. Sci. Eindhoven University of Technology, (1994).
- A. Chalabi, Stable upwind schemes for hyperbolic conservation laws with source terms, IMA J. Numer. Anal. 12 (1992), no. 2, 217–241. MR 1164582, DOI 10.1093/imanum/12.2.217
- 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, DOI 10.1002/cpa.3160470602
- Phillip Colella, Andrew Majda, and Victor Roytburd, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Statist. Comput. 7 (1986), no. 4, 1059–1080. MR 857783, DOI 10.1137/0907073
- Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI 10.1090/S0025-5718-1980-0551288-3
- B. Engquist and B. Sjogreen, Robust difference approximations of stiff inviscid detonation waves, SIAM J. Sci. Comput., to appear.
- Jonathan B. Goodman and Randall J. LeVeque, On the accuracy of stable schemes for $2$D scalar conservation laws, Math. Comp. 45 (1985), no. 171, 15–21. MR 790641, DOI 10.1090/S0025-5718-1985-0790641-4
- Shi Jin, Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 122 (1995), no. 1, 51–67. MR 1358521, DOI 10.1006/jcph.1995.1196
- S. Jin and C. D. Levermore, Numerical schemes for hyperbolic systems with stiff relaxation terms , J. Comput. Phys., Preprint.
- S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
- R. J. LeVeque and H. C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys. 86 (1990), no. 1, 187–210. MR 1033905, DOI 10.1016/0021-9991(90)90097-K
- Tai-Ping Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), no. 1, 153–175. MR 872145, DOI 10.1007/BF01210707
- Andrew Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math. 41 (1981), no. 1, 70–93. MR 622874, DOI 10.1137/0141006
- Stanley Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), no. 2, 217–235. MR 736327, DOI 10.1137/0721016
- R. B. Pemper, Numerical methods for hyperbolic conservation with stiff II., SIAM J. Sci. Comput., 14, (1993), pp. 824-859.
- Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435, DOI 10.1090/S0025-5718-1983-0679435-6
- H. J. Schroll, A. Tveito and R. Winther, An error bound for finite difference schemes applied to a stiff system of conservation laws, Preprint 1994-3, Dept of Informatics, University of Oslo.
- H. J. Schroll and R. Winther, Finite difference schemes for conservation laws with source terms, IMA J. Numer. Anal., 16, (1996), pp. 201-215.
- Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471. MR 954915, DOI 10.1016/0021-9991(88)90177-5
- Eitan Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984), no. 168, 369–381. MR 758189, DOI 10.1090/S0025-5718-1984-0758189-X
- T. Tang and Zhen Huan Teng, Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal. 32 (1995), no. 1, 110–127. MR 1313707, DOI 10.1137/0732004
- B. Van Leer, Towards the ultimate conservative difference schemes V. A second order sequal to Godunov’s method, J. Comput. Phys., 32, (1979), pp. 101-136.
- Bram van Leer, On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Statist. Comput. 5 (1984), no. 1, 1–20. MR 731878, DOI 10.1137/0905001
- J. P. Vila, Convergence and error estimates in finite volume schemes for multidimensional scalar conservation laws; II Implicit monotone schemes, Preprint (1995).
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
Additional Information
- Abdallah Chalabi
- Affiliation: CNRS-UMR MIP 5640 - UFR MIG Universite P. Sabatier, 118, route de Narbonne 31062 Toulouse cedex France
- Email: chalabi@mip.ups-tlse.fr
- Received by editor(s): September 19, 1995
- Received by editor(s) in revised form: March 29, 1996
- © Copyright 1997 American Mathematical Society
- Journal: Math. Comp. 66 (1997), 527-545
- MSC (1991): Primary 35L65, 65M05, 65M10
- DOI: https://doi.org/10.1090/S0025-5718-97-00817-X
- MathSciNet review: 1397441