Diiusive Relaxation Schemes for Discrete-velocity Kinetic Equations

Many kinetic models of the Boltzmann equation have a diiusive scaling that leads to the Navier-Stokes type parabolic equations, such as the heat equation, the porous media equations, the advection-diiusion equation and the viscous Burgers equation. In such problems the diiusive relaxation parameter may diier in several orders of magnitude from the rareeed regimes to the hydrodynamic (diiusive) regimes, and it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter. Earlier approaches that work from the rareeed regimes to the Euler regimes do not directly apply to these problems since here, in addition to the stii relaxation term, the convection term is also stii. Our idea is to reformulate the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stii component of the convection terms into the relaxation term. This, however, introduces new diiculties due to the dependence of the stii source term on the gradient. We show how to overcome this new diiculty with a adequately designed, economical discretization procedure for the relaxation term. These schemes are shown to have the correct diiusion limit. Several numerical results in one and two dimensions are presented, which show the robustness, as well as the uniform accuracy of our schemes.