DIFFUSIVE RELAXATION SCHEMES FOR MULTISCALE DISCRETE-VELOCITY KINETIC-EQUATIONS

Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the Navier-Stokes-type parabolic equations, such as the heat equation, the porous media equations, the advection-diffusion eq uation, and the viscous Burgers equation. In such problems the diffusi ve relaxation parameter may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes, and it is desirable to develop a class of numerical schemes that can work uni formly with respect to this relaxation parameter. Earlier approaches t hat work from the rarefied regimes to the Euler regimes do not directl y apply to these problems since here, in addition to the stiff relaxat ion term, the convection term is also stiff. Our idea is to reformulat e the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stiff component of the con vection terms into the relaxation term. This, however, introduces new difficulties due to the dependence of the stiff source term on the gra dient. We show how to overcome this new difficulty with an adequately designed, economical discretization procedure for the relaxation term. These schemes are shown to have the correct diffusion limit. Several numerical results in one and two dimensions are presented, which show the robustness, as well as the uniform accuracy, of our schemes.