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Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms
Author(s):
A.
Chalabi.
Journal:
Math. Comp.
68
(1999),
955-970.
MSC (1991):
Primary 35L65, 65M05, 65M10
Posted:
February 10, 1999
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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:
- 1.
- D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996), 163-193. CMP 98:13
- 2.
- 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).
- 3.
- A. Bermudez and M. E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids, 23 (1994), pp. 955-970. MR 95i:76065
- 4.
- A. Chalabi, Stable upwind schemes for hyperbolic conservation laws with source terms, IMA J. Numer. Anal., 12 (1992), pp. 955-970. MR 93c:65108
- 5.
- A. Chalabi, An error bound for the polygonal approximation of conservation laws with source terms, Comput. & Math. Appl., 32 (1996), pp. 955-970. MR 97i:65143
- 6.
- A. Chalabi, On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms, Math. Comp., 66 (1997), pp. 527-545. MR 97g:65178
- 7.
- G. Q. Chen, C. D. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), pp. 787-830. MR 95h:35133
- 8.
- P. Colella, A. Majda and V. Roytburd, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 955-970. MR 87i:76037
- 9.
- 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), 400-409. MR 97d:35132
- 10.
- M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation conservation laws, Math. Comp., 34 (1980), pp. 955-970. MR 81b:65079
- 11.
- B. Engquist and B. Sjogreen, Robust difference approximations of stiff inviscid detonation waves, CAM report 91-03, UCLA, Los Angeles, CA, 1991.
- 12.
- J. B. Goodman and R. Leveque, On the accuracy of stable schemes for 2D scalar conservation laws, Math. Comp., 45 (1985), pp. 955-970. MR 86f:65149
- 13.
- S. Jin, Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 122 (1995), pp. 51-67. MR 96g:65084
- 14.
- S. Jin, A convex entropy for a hyperbolic system with relaxation, J. Diff. Equations, 127 (1996), pp. 955-970. MR 97c:35130
- 15.
- S. Jin and C. D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 126 (1996), pp. 955-970. MR 97g:65173
- 16.
- S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), pp. 955-970. MR 96c:65134
- 17.
- P. Klingenstein, Hyperbolic conservation laws with source terms: Errors of the shock location, PhD thesis, Suiss Federal Institute of Tecnology, Zürich (1997).
- 18.
- S. N. Kruzkov, First order quasi-linear equations in several independent variables, Math. USSR-Sb., 81 (1970), pp. 955-970. MR 42:2159
- 19.
- J. LeVeque and H. C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys., 86 (1990), pp. 187-210. MR 90k:76009
- 20.
- T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), pp. 955-970. MR 88f:35092
- 21.
- A. Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math., 40 (1981), pp. 955-970. MR 82j:35096
- 22.
- R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), pp. 955-970. MR 97c:35131
- 23.
- S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal., 21 (1984), pp. 955-970. MR 86d:65119
- 24.
- R. B. Pember, Numerical methods for hyperbolic conservation laws with stiff relaxation II. Higher-order Godunov methods, SIAM J. Sci. Comput., 14 (1993), pp. 955-970. MR 95f:65190
- 25.
- H. J. Schroll, A. Tveito and R. Winther, An
 error bound for semi-implicit difference scheme applied to a stiff system of conservation laws, SIAM J. Numer. Anal. 34 (1997), 1152-1166. MR 98g:65078 - 26.
- H. J. Schroll and R. Winther, Finite difference schemes for conservation laws with source terms, IMA J. Numer. Anal., 16 (1996), pp. 955-970. MR 97g:65177
- 27.
- P. R. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984), pp. 995-1011. MR 85m:65085
- 28.
- E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984), pp. 369-381. MR 86g:65163
- 29.
- T. Tang and Z. H. Teng, Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal., 32 (1995), pp. 955-970. MR 95m:65155
- 30.
- B. Van Leer, Towards the ultimate conservative difference schemes V. A second order sequel to Godunov's method, J. Comput. Phys., 32 (1979), pp. 955-970. CMP 98:05
- 31.
- J. P. Vila, Convergence and error estimates in finite volume schemes for multidimensional scalar conservation laws, Preprint, Nice University (1995).
- 32.
- G. B. Whitham, Linear and nonlinear waves, J. Wiley (1974). MR 58:3905
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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
DOI:
10.1090/S0025-5718-99-01089-3
PII:
S 0025-5718(99)01089-3
Keywords:
Conservation laws,
stiff source term,
relaxation scheme,
fully implicit scheme,
semi-implicit scheme,
TVD scheme,
MUSCL method,
entropy solution
Received by editor(s):
April 29, 1997
Received by editor(s) in revised form:
October 14, 1997
Posted:
February 10, 1999
Copyright of article:
Copyright
1999,
American Mathematical Society
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