Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit
For computational scientists solving hyperbolic systems with stiff relaxation, this work provides a novel numerical method that avoids restrictive time-step constraints in the diffusion limit.
This paper develops Implicit-Explicit Runge-Kutta schemes for hyperbolic systems with stiff relaxation that, in the diffusion limit, yield an IMEX scheme for the convection-diffusion equation where diffusion is treated implicitly, overcoming the parabolic stability restriction. Numerical examples, including neutron transport equations, validate the theoretical analysis.
We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of the paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of the paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by an original reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis on such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation. Several numerical examples including neutron transport equations confirm the theoretical analysis.