  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 procedure. The former preserves the accuracy of the RK schemes and
the latter guarantees the stability. Our method is different from the widely
used approach for the explicit RK schemes by imposing boundary conditions at
intermediate stages, which could not be derived for the IMEX schemes. In
addition, the intermediate boundary conditions are only available for explicit
RK schemes up to third order while our method applies to arbitrary order IMEX
and explicit RK schemes. Moreover, the present boundary treatment method may be
adapted to IMEX RK schemes solving many other partial differential equations.
For a specific third-order IMEX scheme, we demonstrate the good stability and
third-order accuracy of our boundary treatment through both 1D examples and 2D
reactive Euler equations.
