Discontinuous Galerkin schemes for multi-dimensional coupled hyperbolic systems

TL;DR

The authors address multi-dimensional coupled hyperbolic systems separated by a sharp interface and introduce a relaxation-based DG framework that sidesteps nonlinear half-Riemann problems. By embedding the original problem into a Jin-Xin relaxation with a local normal projection, they derive a consistent relaxed coupling condition and construct problem-specific Riemann solvers. The resulting DG-IMEX schemes are high-order, SSP-stable, and asymptotically preserve the relaxation limit, yielding accurate, interface-resolved solutions. The approach is validated on a fluid–structure interaction example, demonstrating complex wave phenomena and adherence to physical coupling constraints while achieving high resolution with adaptive mesh refinement. Overall, the work provides a flexible, plug-and-play method for a broad class of multi-dimensional coupled hyperbolic systems with interfaces.

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.

Figures (3)

• Figure 1: Domains of the model problem and the quasi-one-dimensional coupling problem in $\mathbb{R}^2$. While in the model problem two systems of conservation laws are coupled on a general interface the quasi-one-dimensional coupling originates from a localization of the problem at $\hat{\mathbf{x}} \in \Gamma$.
• Figure 2: The coupled half-Riemann problem for the localized relaxation system in the $x$-$t$-plane. The left trace data $\mathbf{Q}^{-}$ are connected to the coupling data $\mathbf{Q}_R$ by Lax-curves of negative speeds, i.e., $\mathcal{L}_{1}^1\dots\mathcal{L}_{1}^{m_1}$ and the outgoing trace data $\mathbf{Q}^{+}$ are connected to the coupling data $\mathbf{Q}_L$ by Lax-curves of positive speeds, i.e., $\mathcal{L}_{2}^{m_2+1}\dots \mathcal{L}_2^{2m_1}$. Coupling data at the interface are related by the coupling condition $\Psi_Q^\mathbf{n}$.
• Figure 3: Numerical solution of the fluid-structure coupling problem in terms of negative stress $-\sigma_{11}$ in the solid domain ($x_1<0$ m) and pressure $p$ in the fluid domain ($x_1>0$ m) over six time instances.

Theorems & Definitions (17)

• Remark 2.1
• Remark 2.2
• Definition 2.1
• Definition 3.1
• Definition 3.2
• Proposition 3.2
• proof
• Remark 4.1
• Definition 4.1
• Lemma 4.2