Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms
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Abstract:
We focus in this study on the convergence of a class of relaxation numerical schemes for hyperbolic scalar conservation laws including stiff source terms. Following Jin and Xin, we use as approximation of the scalar conservation law, a semi-linear hyperbolic system with a second stiff source term. This allows us to avoid the use of a Riemann solver in the construction of the numerical schemes. The convergence of the approximate solution toward a weak solution is established in the cases of first and second order accurate MUSCL relaxed methods.References
- D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996), 163–193.
- 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, Eindhoven University of Technology, (1994).
- Alfredo Bermudez and Ma. Elena Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids 23 (1994), no. 8, 1049–1071. MR 1314237, DOI 10.1016/0045-7930(94)90004-3
- 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
- A. Chalabi, An error bound for the polygonal approximation of conservation laws with source terms, Comput. Math. Appl. 32 (1996), no. 8, 59–63. MR 1426198, DOI 10.1016/0898-1221(96)00167-8
- Abdallah Chalabi, On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms, Math. Comp. 66 (1997), no. 218, 527–545. MR 1397441, DOI 10.1090/S0025-5718-97-00817-X
- 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
- J. F. Collet and M. Rascle, Convergence of the relaxation approximation to a scalar nonlinear hyperbolic equation arising in chromatography, Z. Angew. Math. Phys. 47 (1996), no. 3, 400–409. MR 1394915, DOI 10.1007/BF00916646
- 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, CAM report 91-03, UCLA, Los Angeles, CA, 1991.
- 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
- Shi Jin, A convex entropy for a hyperbolic system with relaxation, J. Differential Equations 127 (1996), no. 1, 97–109. MR 1387259, DOI 10.1006/jdeq.1996.0063
- Shi Jin and C. David Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 126 (1996), no. 2, 449–467. MR 1404381, DOI 10.1006/jcph.1996.0149
- 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 10.1002/cpa.3160480303
- P. Klingenstein, Hyperbolic conservation laws with source terms: Errors of the shock location, PhD thesis, Suiss Federal Institute of Tecnology, Zürich (1997).
- 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
- 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 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3
- 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
- Richard B. Pember, Numerical methods for hyperbolic conservation laws with stiff relaxation. II. Higher-order Godunov methods, SIAM J. Sci. Comput. 14 (1993), no. 4, 824–859. MR 1223276, DOI 10.1137/0914052
- Hans Joachim Schroll, Aslak Tveito, and Ragnar Winther, An $L^1$-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, DOI 10.1137/S0036142994268855
- Hans Joachim Schroll and Ragnar Winther, Finite-difference schemes for scalar conservation laws with source terms, IMA J. Numer. Anal. 16 (1996), no. 2, 201–215. MR 1382716, DOI 10.1093/imanum/16.2.201
- P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984), no. 5, 995–1011. MR 760628, DOI 10.1137/0721062
- 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 sequel to Godunov’s method, J. Comput. Phys., 32 (1979), pp. 101-136.
- J. P. Vila, Convergence and error estimates in finite volume schemes for multidimensional scalar conservation laws, Preprint, Nice University (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
- A. Chalabi
- Affiliation: CNRS, Umr Mip 5640 - UFR Mig Universite P. Sabatier, Route de Narbonne 31062 Toulouse cedex France
- Email: chalabi@mip.ups-tlse.fr
- Received by editor(s): April 29, 1997
- Received by editor(s) in revised form: October 14, 1997
- Published electronically: February 10, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 955-970
- MSC (1991): Primary 35L65, 65M05, 65M10
- DOI: https://doi.org/10.1090/S0025-5718-99-01089-3
- MathSciNet review: 1648367