Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation

We consider implicit-explicit (IMEX) Runge Kutta methods for
hyperbolic systems of conservation laws with stiff relaxation
terms. Here we restrict our analysis to diagonally implicit Runge
Kutta (DIRK) schemes and consider schemes that are asymptotic
preserving (AP) and strong-stability-preserving (SSP) in the zero
relaxation limit. Accuracy and stability properties of these
schemes are studied both analytically and numerically. The IMEX
schemes are then combined with a weighted essentially non
oscillatory (WENO) finite difference strategy in space to obtain
fully discrete high order schemes. Several applications to
hyperbolic relaxation systems in stiff and non stiff regimes show
the efficiency and the robustness of the present approach.