An Asymptotic Preserving Two-Dimensional Staggered Grid Method for multiscale transport equations

We propose a two-dimensional asymptotic preserving scheme for linear transport equations with diffusive scalings. It is an extension of the time splitting developed by Jin, Pareschi and Toscani [SINUM,2000], but uses spatial discretizations on staggered grids, which preserves the discrete diffusion limit with a more compact stencil. The first novelty of this paper is that we propose a staggering in two dimensions that requires fewer unknowns than one could have naively expected. The second contribution of this paper is that we rigorously analyze the scheme of Jin, Pareschi, and Toscani [SINUM,2000] We show that the scheme is AP and obtain an explicit CFL condition, which couples a hyperbolic and a parabolic condition. This type of condition is common for asymptotic preserving 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.