An Asymptotic Preserving 2--D Staggered Grid Method for multiscale transport equations
Kerstin Küpper, Martin Frank, Shi Jin · arXiv (Cornell University) · 2015
We propose a two-dimensional asymptotic-preserving scheme for linear transport equations with diffusive scalings. It is based on the time splitting developed by Jin, Pareschi and Toscani [SINUM, 2000], but takes spatial discretizations on staggered grids. Compared with the previous methods based on regular Cartesian grids, this method preserves the discrete diffusion limit with a more compact stencil thus has a better spatial resolution. This scheme requires less unknowns than one could have naively expected for a method based on staggered grids. We show that the scheme is AP and we provide a stability analysis to obtain an explicit CFL condition, which couples a hyperbolic and a parabolic condition. This type of condition is common for AP schemes and guarantees uniform stability with respect to the mean free path. In addition, we obtain an upper bound on the relaxation parameter, which is the crucial parameter of the used time discretization. Several numerical examples are provided to verify the accuracy and asymptotic property of the scheme.