Abstract

Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time $\varepsilon$). To solve this type of problems, implicit-explicit (IMEX) methods have been widely used and their performance is understood well in the non-stiff regime ($\varepsilon=O(1)$) and limiting regime ($\varepsilon\rightarrow0$). However, in the intermediate regime (say, $\varepsilon=O(\Delta t)$), uniform accuracy has been reported numerically for most IMEX multistep methods, while complicated behavior of order reduction has been observed for IMEX Runge-Kutta (RK) methods. In this talk, I will take a linear hyperbolic systems with stiff relaxation as a model problem, and discuss how to use energy estimates with multiplier techniques to prove the uniform accuracy of IMEX methods. In particular, I will present my joint works with Jingwei Hu on the uniform accuracy of IMEX backward differentiation formulas (IMEX-BDF) up to fourth order, and that of IMEX -RK methods up to third order.