Abstract : When designing high order schemes for solving time-dependent kinetic and related PDEs, we often first develop semi-discrete schemes paying attention only to spatial discretizations and leaving time $t$ continuous. It is then important to have a high order time discretization to main the stability properties of the semi-discrete schemes. In this talk we discuss two classes of high order time discretization, i.e, the strong stability preserving (SSP) time discretization, which preserves strong stability from a stable spatial discretization with Euler forward, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes. Numerical examples will be given to demonstrate the performance of these schemes.

Keywords : time discretization; strong stability; Runge-Kutta; multistep methods