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Numerical schemes for kinetic equations in diffusive regimes. (English) Zbl 1337.65118

Summary: The diffusive scaling of many finite-velocity kinetic models leads to a small-relaxation time behavior governed by reduced systems which are parabolic in nature. Here we demonstrate that standard numerical methods for hyperbolic conservation laws with stiff relaxation fail to capture the right asymptotic behavior. We show how to design numerical schemes for the study of the diffusive limit that possess the discrete analogue of the continuous asymptotic limit. Numerical results for a model of relaxing heat flow and for a model of nonlinear diffusion are presented.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] Carleman, T., Problémes mathèmatiques dans la thèorie cinètique des gas, Publ. Scient. Inst. Mittag-Leffler (1957) · Zbl 0077.23401
[2] Kurz, T. G., Convergence of sequence of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc., 186, 259-272 (1973)
[3] McKean, H. P., The central limit theorem for Carleman’s equation, Israel J. of Math., 21, 54-92 (1975) · Zbl 0315.60013
[4] P.L. Lions and G. Toscani, Diffusive limit for two-velocity Boltzmann kinetic models, Rev. Math. Iberoamericana; P.L. Lions and G. Toscani, Diffusive limit for two-velocity Boltzmann kinetic models, Rev. Math. Iberoamericana · Zbl 0896.35109
[5] Jin, S.; Levermore, D., Numerical schemes for hyperbolic conservation laws with stiff relaxation, J. Comp. Phys., 126, 449-467 (1996) · Zbl 0860.65089
[6] Pember, R. B., Numerical methods for hyperbolic conservation laws with stiff relaxation, I. Spurious solutions, SIAM J. Appl. Math., 53, 1293-1330 (1993) · Zbl 0787.65062
[7] Caflish, R. E.; Jin, S.; Russo, G., Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Num. Anal., 34, 1, 246-281 (1997) · Zbl 0868.35070
[8] E. Gabetta, L. Pareschi and G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Num. Anal.; E. Gabetta, L. Pareschi and G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Num. Anal. · Zbl 0897.76071
[9] Roe, P. L.; Arora, M., Characteristic-based schemes for dispersive waves I. The method of characteristics for smooth solutions, Numerical Meth. for PDE’s, 9, 459-505 (1993) · Zbl 0787.65066
[10] S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for discrete-velocity kinetic equations, SIAM J. Num. Anal.; S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for discrete-velocity kinetic equations, SIAM J. Num. Anal. · Zbl 0938.35097
[11] Naldi, G.; Pareschi, L., Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation (1997), (preprint)
[12] Barenblatt, G. I., On some unsteady motion of a liquid or a gas in a porous medium, Prikl. Math. Meh., 16, 67 (1952) · Zbl 0049.41902
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