An asymptotic preserving two-dimensional staggered grid method for multiscale transport equations (Q2788630)

A two-dimensional asymptotic-preserving (AP) scheme to solve linear transport equations is presented. The considered linear transport equation has the diffusion equation as an analytical asymptotic limit. In the case of AP schemes, the discretization has to be chosen such that the analytic limit is preserved at a discrete level and the scheme is uniformly stable with respect to the mean free path. The present work is an extension of the time splitting scheme developed by Jin, Pareschi, and Toscani [\textit{S. Jin} et al., ibid. 38, No. 3, 913--936 (2000; Zbl 0976.65091)], but it uses spatial discretizations on staggered grids, which preserves the discrete diffusion limit with a more compact stencil.NEWLINENEWLINEThe first novelty of the paper is that a staggering in two dimensions is proposed, that requires fewer unknowns than one could have expected. The second contribution of this paper is that it rigorously analyzes the scheme of Jin et al. [loc. cit.]. It is shown that the scheme is AP, and an explicit so-called CFL (Courant-Friedrich-Lewy) condition is obtained, which couples a hyperbolic and a parabolic condition. This type of CFL-condition is common for asymptotic-preserving schemes and guarantees uniform stability with respect to the mean free path. In addition, an upper bound on the relaxation parameter is found, which is the crucial parameter of the used time discretization. Several numerical tests are provided to verify the accuracy and the asymptotic properties of the two-dimensional scheme. Here, the order of convergence of the scheme with respect to the spatial variable is examined, the method of manufactured solutions is applied, a Gauss test is performed, an example with space-dependent scattering is considered, and a two-material test is done.