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

## TL;DR

This paper introduces a high-order finite difference boundary treatment for IMEX Runge-Kutta schemes solving hyperbolic systems with source terms, ensuring accuracy and stability on Cartesian meshes.

## Contribution

It develops a novel boundary treatment combining RK schemes at the boundary with an inverse Lax-Wendroff procedure, applicable to arbitrary order IMEX schemes.

## Key findings

- Achieves third-order accuracy in boundary treatment.
- Demonstrates stability and accuracy on 1D and 2D examples.
- Applicable to various PDEs with IMEX RK schemes.

## 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 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.