Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

Abstract

In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme. We address this problem by combining the idea of using the RK schemes at the boundary and an inverse Lax-Wendroff (ILW) procedure in [29,30]. Specifically, we only apply the ILW procedure in the starting stage of the RK method. In the intermediate stages, we solve out the intermediate solutions as well as their first-order spatial derivatives at the boundary by using the RK schemes, which are then used to compute solutions at ghost points by Taylor expansions. Our method is different from the widely used approach for the explicit RK schemes by imposing boundary conditions at intermediate stages [14,29,30] which does not apply to IMEX schemes. In addition, the intermediate boundary conditions are available for explicit RK schemes only up to third order while our method applies to IMEX and explicit RK schemes of arbitrary order. The stability and the desired accuracy of our boundary treatment with IMEX RK method up to fourth order are demonstrated numerically through 1D examples and 2D reactive Euler equations.