We propose 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 low-rank approximation is only used for the micro part of the solution. The time and spatial discretizations are done properly so that the overall scheme is second-order accurate (in both the fully kinetic and the limit regime) and asymptotic-preserving (AP). That is, in the diffusive regime, the scheme becomes a macroscopic solver for the limiting diffusion equation that automatically captures the low-rank structure of the solution. Moreover, the method can be implemented in a fully explicit way and is thus significantly more efficient compared to the previous state of the art. We demonstrate the accuracy and efficiency of the proposed low-rank method by a number of four-dimensional (two dimensions in physical space and two dimensions in velocity space) simulations.

我们提出了一种动态低秩方法来降低求解多维多维线性运输方程的计算复杂度。该方法基于方程的宏观-微观分解。低秩近似仅用于解决方案的微小部分。适当地进行时间和空间离散化，以便整体方案是二阶精确的（在完全动力学和极限状态下）和渐近保持（AP）。就是说，在扩散状态下，该方案成为了极限扩散方程的宏观求解器，该方程自动捕获了溶液的低阶结构。此外，该方法可以以完全明确的方式实现，因此与现有技术水平相比，效率显着提高。