Relaxation schemes for nonlinear kinetic equations

SIAM Journal on Numerical Analysis

Volumn 34, Issue 6, 1997, Pages 2168-2194

  Gabetta, E ^a   Pareschi, L ^b   Toscani, G ^a  

^a UNIVERSITY OF PAVIA   (Italy)

^b UNIVERSITY OF FERRARA   (Italy)

Author keywords

Boltzmann equation; Fluid dynamic limit; Wild sum

Indexed keywords

EID: 0000677613     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142995287768     Document Type: Article

References (32)

1. L. ARKERYD, On the Boltzmann equation I and II, Arch. Rational Mech. Anal., 45 (1972), pp. 1-34.
2. H. BABOVSKY, On a simulation scheme for the Boltzmann equation, Math. Meth. Appl. Sci., 8 (1986), pp. 223-233.
3. H. BABOVSKY AND R. ILLNER, A convergence proof for Nambu's simulation method for the full Boltzmann equation. SIAM J. Numer. Anal., 26 (1989), pp. 45-65.
4. G.A. BIRD, Molecular Gas Dynamics and Direct Simulation of Gas Flows, Clarendon Press, Oxford, UK, 1994.
5. A.V. BOBYLEV, E. GABETTA, AND L. PARESCHI, On a boundary value problem for the plane Broadwell model. Exact solutions and numerical simulation, Math. Models Methods Appl. Sci., 5 (1995). pp. 253-266.
6. A.V. BOBYLEV AND G. TOSCANI. On the generalization of the Boltzmann H-theorem for a spatially homogeneous Maxwell gas, J. Math. Phys., 33 (1992), pp. 2578-2586.
7. J.E. BROADWELL, Shock structure in a simple discrete velocity gas, Phys. Fluids, 7 (1964), pp. 1243-1247.
8. Y. BRENIER, Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), pp. 1013-1037.
9. R. CAFLISCH, S. JIN. AND G. RUSSO, Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal., 34 (1997), pp. 246-281.
10. C. CERCIGNANI, Sur des critères d'existence glôbale en théorie cinétique discrète, C. R. Acad. Sci. Paris I, 301 (1985), pp. 89-92.
11. C. CERCIGNANI, The Boltzmann Equation and Its Applications, Springer-Verlag. New York, 1988.
12. F. CORON AND B. PERTHAME, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), pp. 26-42.
13. L. DESVILLETTES AND S. MISCHLER, About the splitting algorithm, for Boltzmann and BGK equations, Math. Model Methods Appl. Sci., 6 (1996), pp. 1079-1101.
14. R. DIPERNA AND P.L. LIONS. On the Cauchy problem for Boltzmann equations: Global exis- tence and weak stability. Ann. Math., 130 (1989), pp. 321-366.
15. E. GABETTA AN L. PARESCHI, Approximating the Broadwell model in a strip, Math. Models Methods Appl. Sci., 2 (1992), pp. 1-19.
16. R. GATIGNOL, Thèorie cinétique d'un gas à répartition discrète de vitesses, Lecture Notes in Phys., 36, Springer-Verlag, Heidelberg, 1975.
17. J.V. HEROD, Series solutions for nonlinear Boltzmann equations, Nonlinear Anal., 7 (1983), pp. 1373-1387.
18. T. INAMURO AND B. STURTEVANT, Numerical study of discrete velocity gases, Phys. Fluids A, 2 (1990), pp. 2196-2203.
19. R. ILLNER AND H. NEUNZERT, On simulation methods for the Boltzmann equation, Transport Theory Statist. Phys., 16 (1987), pp. 141-154.
20. S. JIN, Runge-Kutta methods for hyperbolic, conservation laws with stiff relaxation terms, J. Comp. Phys., 122 (1995), pp. 51-67.
21. Z. KIELEK, An application of the convolution iterates to evolution equations in Banach space, Univ. lagel. Acta Math., XXVII, (1988).
22. R.J. LEVEQUE, Numerical Methods for Conservation Laws, Birkhauser-Verlag, Basel, 1992.
23. K. NAMBU, Direct simulation scheme derived from the Boltzmann equation, J. Phys. Soc. Japan, 49 (1980), pp. 2042-2049.
24. L. PARESCHI AND B. PERTHAME, A Fourier spectral method for homogeneous Boltzmann equations, Transport Theory Statist. Phys., 25 (1996), pp. 369-383.
25. R.B. PEMBER, Numerical methods for hyperbolic conservation laws with stiff relaxation, I, Spurious solutions, SIAM J. Appl. Math., 53 (1993), pp. 1293-1330.
26. R.B. PEMBER, Numerical methods for hyperbolic conservation laws with stiff relaxation, II, High order Goudonov methods, SIAM J. Sci. Stat. Comput., 14 (1993), pp. 824-859.
27. B. PERTHAME AND Y. Qiu, A variant of Van Leer's method for multidimensional system of conservation laws, J. Comp. Phys., 1/2 (1994), pp. 370-381.
28. A. PEYERIMHOFF, Lecture Notes on Summability, Lecture Notes in Math. 107, Springer-Verlag. Heidelberg, 1969.
29. P.L. ROE AND D. SIDILKOVER, Optimum positive linear schemes for advection in two and three dimensions, SIAM J. Numer. Anal., 29 (1992), pp. 1542-1568.
30. F. ROGIER AND J. SCHNEIDER, A direct method for solving the Boltzmann equation, Transport Theory Statist. Phys., 23 (1994), pp. 313-338.
31. G. TOSCANI AND W. WALUS, The initial-boundary value problem for the four velocity plane Broadwell model, Math. Models Methods Appl. Sci., 1 (1991). pp. 293-310.
32. E. WILD, On Boltzmann's equation in the kinetic theory of gases, Proc. Camb. Philos. Soc., 47 (1951), pp. 602-609.