Time-splitting schemes and measure source terms for a quasilinear relaxing system

Mathematical Models and Methods in Applied Sciences

Volumn 13, Issue 8, 2003, Pages 1081-1101

  Gosse, Laurent ^a  

^a CNR   (Italy)

Author keywords

Conservation laws; Nonconservative products; Relaxation scheme

Indexed keywords

EID: 0242474571     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202503002829     Document Type: Article

References (34)

1. D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 61 (1996) 163-196.
2. F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Stat. Phys. 95 (1999) 113-170.
3. F. Bouchut and B. Perthame, Kružkov's inequalities for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998) 2847-2870.
4. H. Brezis, Analyse Fonctionnelle: Théorie et Applications (Masson, 1983).
5. J. F. Colombeau and A. Heibig, Nonconservative products in bounded variation functions, SIAM J. Math. Anal. 23 (1992) 941-949.
6. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften, Vol. 325 (Springer-Verlag, 1999).
7. L. Gosse, Localization effects and measure source terms in numerical schemes for balance laws, Math. Comput. 71 (2002) 553-582.
8. L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C.R. Acad. Sci. Paris 334 (2002) 337-342.
9. L. Gosse and A. E. Tzavaras, Convergence of relaxation approximations to the equations of elastodynamics, Math. Comput. 70 (2001) 555-577.
10. C. Helzel, R. J. LeVeque and G. Warnecke, A modified fractional step method for the accurate approximation of detonation waves, SIAM J. Sci. Comput. 22 (2000) 1489-1510.
11. F. James, Convergence results for some conservation laws with a reflux boundary condition and a relaxation term arising in chemical engineering, SIAM J. Math. Anal. 29 (1998) 1200-1223.
12. S. Jin, Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 122 (1995) 51-67.
13. S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal. 35 (1998) 2405-2439.
14. S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal. 38 (2000) 913-936.
15. S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235-277.
16. M. A. Katsoulakis, G. T. Kossioris and Ch. Makridakis, Convergence and error estimates of relaxation schemes for multidimensional conservation laws, Comm. Partial Differential Equations 24 (1999) 395-424.
17. M. A. Katsoulakis and A. E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations 22 (1997) 195-233.
18. S. N. Kružkov, First order quasilinear equations in several independant space variables, Math. USSR Sbornik 10 (1970) 217-243.
19. A. Kurganov and E. Tadmor, Stiff systems of hyperbolic conservation laws: convergence and error estimates, SIAM J. Math. Anal. 28 (1997) 1446-1456.
20. N. N. Kuznečov, The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation (Russian), Ž. Vyčisl, Mat. Mat. Fiz. 16 (1976) 1489-1502.
21. P. G. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of non-conservative products, SIAM J. Math. Anal. 30 (1999) 1309-1342.
22. P. L. Lions and G. Toscani, Diffusive limits for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana 13 (1997) 1-34.
23. T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987) 153-175.
24. C. Klingenberg and Y. G. Lu, Cauchy problem for hyperbolic conservation laws with a relaxation term, Proc. Roy. Soc. Edinburgh Sec. A 126 (1996) 821-828.
25. G. Naldi and L. Pareschi, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation, SIAM J. Numer. Anal. 37 (2000) 1246-1270.
26. R. Natalini, Recent results on hyperbolic relaxation problems, in Analysis of Systems of Conservation Laws, Aachen, 1997, Monogr. Surv. Pure Appl. Math., Vol. 99 (Chapman & Hall/CRC, 1999).
27. R. Pember, Numerical methods for hyperbolic conservation laws with stiff relaxation I. Spurious solutions, SIAM J. Appl. Math. 53 (1993) 1293-1330.
28. B. Perthame and E. Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys. 138 (1991) 501-517.
29. J. P. Raymond, A new definition of nonconservative products and weak stability results, Boll. Un. Mat. Ital. B 10 (1996) 681-699.
30. K. Rhee, R. Aris and N. R. Amundsen, First-Order Partial Differential Equations I & II, Prentice Hall International Series Vol. 1986 (Prentice Hall, 1989).
31. 1 -error bound for a semi-implicit difference scheme applied to a stiff system of conservation laws, SIAM J. Numer. Anal. 34 (1997) 1152-1166.
32. A. Tveito and R. Winther, On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term, SIAM J. Math. Anal. 28 (1997) 136-161.
33. A. Vasseur, Kinetic semi-discretization of scalar conservation laws and convergence by using averaging lemmas, SIAM J. Numer. Anal. 36 (1999) 465-474.
34. A. I. Vol'Pert, The space BV and quasilinear equations, Math. USSR Sbornik 2 (1967) 225-267.