An asymptotic-preserving dynamical low-rank method for the multi-scale multi-dimensional linear transport equation. (English) Zbl 07512318
Summary: 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.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 65Lxx | Numerical methods for ordinary differential equations |
| 82Cxx | Time-dependent statistical mechanics (dynamic and nonequilibrium) |
Keywords:
dynamical low-rank integrator; linear transport equation; macro-micro decomposition; diffusion limit; asymptotic preserving; implicit-explicit Runge-Kutta scheme (IMEX)References:
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