Discontinuous Galerkin schemes for multi-dimensional coupled hyperbolic systems

TL;DR

This paper introduces a new class of high-order discontinuous Galerkin schemes for multi-dimensional coupled hyperbolic systems with sharp interfaces, avoiding complex Riemann problem solutions by using a relaxation approach.

Contribution

The paper presents a novel relaxation-based discontinuous Galerkin scheme for coupled hyperbolic systems that simplifies interface treatment and incorporates high-order time discretization.

Findings

Effective handling of multi-dimensional coupled systems with sharp interfaces.

Avoidance of nonlinear Riemann problem solutions.

Successful application to fluid-structure interaction problems.

Abstract

A novel class of Runge-Kutta discontinuous Galerkin schemes for coupled systems of conservation laws in multiple space dimensions that are separated by a fixed sharp interface is introduced. The schemes are derived from a relaxation approach and a local projection and do not require expensive solutions of nonlinear half-Riemann problems. The underlying Jin-Xin relaxation involves a problem specific modification of the coupling condition at the interface, for which a simple construction algorithm is presented. The schemes are endowed with higher order time discretization by means of strong stability preserving Runge-Kutta methods. These are derived from an asymptotic preserving implicit-explicit treatment of the coupled relaxation system taken to the discrete relaxation limit. In a case study the application to a multi-dimensional fluid-structure coupling problem employing the compressible Euler equations and a linear elastic model is discussed.