An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation

Abstract: We introduce a dynamical low-rank method to reduce the computational complexity for solving the multi-scale multi-dimensional linear transport equation. The method is based on a macro-micro decomposition of the equation. The proposed numerical method uses the low rank approximation only for the micro part of the solution. The time and spatial discretizations are done properly so that the overall scheme is second order accurate and asymptotic-preserving (AP); that is, in the diffusive regime, the scheme becomes a macroscopic solver for the limiting diffusion equation and is automatically low rank. We demonstrate the accuracy and efficiency of the proposed low rank method by a number of two-dimensional examples.