Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes

SIAM Journal on Numerical Analysis

Volumn 41, Issue 2, 2003, Pages 641-658

  Gosse, Laurent ^a   Toscani, Giuseppe ^b  

^a Istituto per le Applicazioni del Calcolo Sezione di Bari   (Italy)

^b UNIVERSITY OF PAVIA   (Italy)

Author keywords

Diffusive relaxation schemes; Kinetic equations; Nonconservative products

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; FUNCTION EVALUATION; MATHEMATICAL MODELS; MATHEMATICAL TRANSFORMATIONS; PROBLEM SOLVING; SET THEORY;

DIFFUSIVE RELAXATION SCHEMES; KINETIC EQUATIONS; NONCONSERVATIVE PRODUCTS; SPACE LOCALIZATIONS;

LINEAR EQUATIONS;

EID: 2342563079     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142901399392     Document Type: Article

References (39)

1. D. AMADORI, L. GOSSE, AND G. GUERRA, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal., 162 (2002), pp. 327-366.
2. D. AREGBA-DRIOLLET, R. NATALINI, AND S.Q. TANG, Diffusive kinetic explicit schemes for nonlinear degenerate parabolic systems, Math. Comp., to appear.
3. G.I. BARENBLATT, On some steady motion of a liquid or a gas in a porous medium, Prikl. Mat. Mekh., 16 (1952), pp. 67-78.
4. R. BOTSCHORIJVILI, B. PERTHAME, AND A. VASSEUR, Equilibrium schemes for scalar conservation laws with stiff source terms, Math. Comp., 72 (2003), pp. 131-157.
5. F. BOUCHUT, F.R. GUARGUAGLINI, AND R. NATALINI, Diffusive BGK approximations for nonlinear multidimensional parabolic equations, Indiana Univ. Math. J., 49 (2000), pp. 723-749.
6. T. CARLEMAN, Problèmes mathématiques de la théorie cinétique des gaz, Almqvist and Wiksell, Uppsala, Sweden, 1957.
7. C. CERCIGNANI, R. ILLNER, AND M. PULVIRENTI, The Mathematical Theory of Dilute Gases, Appl. Math. Sci. 106, Springer-Verlag, New York, 1994.
8. E. GABETTA AND B. PERTHAME, Scaling limits for the Ruijgrok-Wu model of the Boltzmann equation, Math. Methods Appl. Sci., 24 (2001), pp. 949-967.
9. S.K. GODUNOV, Finite difference schemes for numerical computation of solutions of the equations of fluid dynamics, Mat. USSR Sbornik, 47 (1959), pp. 271-306.
10. L. GOSSE, Localization effects and measure source terms in numerical schemes for balance laws, Math. Comp., 71 (2002), pp. 553-582.
11. L. GOSSE, Time-splitting schemes and measure source terms for a quasilinear relaxing system, Math. Models Methods Appl. Sci., to appear.
12. L. GOSSE, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms, Math. Models Methods Appl. Sci., 11 (2001), pp. 339-365.
13. L. GOSSE AND G. TOSCANI, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C.R. Math. Acad. Sci. Paris Sér. I Math., 334 (2002), pp. 337-342.
14. L. GOSSE AND A.E. TZAVARAS, Convergence of relaxation schemes to the equations of elasto-dynamics, Math. Comp., 70 (2001), pp. 555-577.
15. J.M. GREENBERG AND A.Y. LERoux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), pp. 1-16.
16. xx, Comm. Pure Appl. Math., 3 (1950), pp. 201-230.
17. S. JIN, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), pp. 441-454.
18. S. JIN, A steady-state capturing method for hyperbolic systems with geometrical source terms, M2AN Math. Model. Numer. Anal., 35 (2001), pp. 631-645.
19. F. GOLSE, S. JIN, AND C.D. LEVERMORE, The convergence of numerical transfer schemes in diffusive regimes I: Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), pp. 1333-1369.
20. S. JIN AND H. LIU, Diffusion limit of a hyperbolic system with relaxation, Methods Appl. Anal., 5 (1998), pp. 317-334.
21. S. JIN, L. PARESCHI, AND G. TOSCANI, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal., 35 (1998), pp. 2405-2439.
22. A. KLAR, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35 (1998), pp. 1073-1094.
23. A. KLAR AND A. UNTERREITER, Uniform stability of a finite difference scheme for transport equations in diffusive regimes, SIAM J. Numer. Anal., 40 (2002), pp. 891-913.
24. S.N. KRUŽKOV, First order quasilinear equations in several independent space variables, Mat. USSR Sbornik, 81 (1970), pp. 228-255.
25. PH. G. LEFLOCH AND A.E. TZAVARAS, Representation of weak limits and definition of non-conservative products, SIAM J. Math. Anal., 30 (1999), pp. 1309-1342.
26. R.J. LEVEQUE, Balancing source terms and flux gradients in high resolution Godunov methods: The quasi steady wave propagation algorithm, J. Comput. Phys., 146 (1998), pp. 346-365.
27. R.J. LEVEQUE AND B. TEMPLE, Stability of Godunov's method for a class of 2 × 2 systems of conservation laws, Trans. Amer. Math. Soc., 288 (1985), pp. 115-123.
28. P.L. LIONS, Mathematical topics in fluid mechanics I, Incompressible models, Oxford University Press, London, 1996.
29. P.L. LIONS AND G. TOSCANI, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana, 13 (1997), pp. 473-513.
30. H. LIU, J. WANG, AND G. WARNECKE, Convergence of a splitting scheme applied to the Ruijgrok-Wu model of the Boltzmann equation, J. Comput. Appl. Math., 134 (2001), pp. 343-367.
31. P. MARCATI AND A. MILANI, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differential Equations, 84 (1990), pp. 129-147.
32. G. NALDI AND L. PARESCHI, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation, SIAM J. Numer. Anal., 37 (2000), pp. 1246-1270.
33. R. NATALINI AND B. HANOUZET, Weakly coupled systems of quasilinear hyperbolic equations, Differential Integral Equations, 9 (1997), pp. 1279-1292.
34. B. PERTHAME AND C. SIMEONI, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), pp. 201-231.
35. B. PERTHAME AND E. TADMOR, A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys., 136 (1991), pp. 501-517.
36. T. PLATKOWSKI AND R. ILLNER, Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory, SIAM Rev., 30 (1988), pp. 213-255.
37. W. RUIJGROK AND T.T. WU, A completely solvable model of the nonlinear Boltzmann equation, Phys. A, 113 (1982), pp. 401-416.
38. A.E. TZAVARAS, On the mathematical theory of fluid dynamic limits to conservation laws, in Advances in Mathematical Fluid Mechanics, J. Malek, J. Nečas, and M. Rokyta, eds., Springer, New York, 2000, pp. 192-222.
39. A.I. VOL'PERT, Spaces BV and quasilinear equations, Mat. USSR Sbornik, 2 (1967), pp. 225-267.