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 equation, and the viscous Burgers equation. In such problems the diffusive 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 uniformly with respect to this relaxation parameter. Earlier approaches that work from the rarefied regimes to the Euler regimes do not directly apply to these problems since here, in addition to the stiff relaxation term, the convection term is also stiff. Our idea is to reformulate the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stiff component of the convection terms into the relaxation term. This, however, introduces new difficulties due to the dependence of the stiff source term on the gradient. 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.