Kinetic transport equations are used to model phenomena such as radiative transfer and neutron transport. In particular, we consider a linear kinetic transport equation under a diffusive scaling that converges to a diffusion equation as the Knudsen number epsilon goes to 0. One challenge that arises when solving this equation is that it is multiscale in epsilon; namely, the equation behaves like a transport or diffusion equation depending on if epsilon = O(1) or epsilon << 1. In addition, the stable time step dt used by standard explicit numerical schemes depends on epsilon which becomes costly when epsilon << 1. Further, our goal was to develop high-order asymptotic preserving (AP) schemes to solve this equation and assess the accuracy and stability of the schemes. Our schemes use implicit-explicit (IMEX)-BDF methods for the temporal discretization, discontinuous Galerkin (DG) methods for the spatial discretization, and the discrete ordinates method for the velocity discretization. Through Fourier analysis, we found that the first, second, and third order in time and space schemes are unconditionally stable when epsilon << 1 and dt = O(epsilon*dx) when epsilon = O(1) where dx is the spatial mesh size. We further applied the third order in time and space scheme to various numerical examples to demonstrate its accuracy.
High-Order Asymptotic Preserving IMEX-BDF-DG Schemes for Linear Kinetic Transport Equation
Kimberly Matsuda, Rensselaer Polytechnic InstituteAuthors: Kimberly Matsuda, Fengyan Li
2023 AWM Research Symposium