Coupling conditions for linear hyperbolic relaxation systems in two-scales problems

TL;DR

This paper develops and validates coupling conditions for linear hyperbolic relaxation systems with multiple relaxation times, enabling efficient domain decomposition and stable numerical solutions using a DG scheme.

Contribution

It derives and proves the validity of coupling conditions for multi-relaxation systems and introduces a stable DG scheme for interface problems.

Findings

Coupling conditions are valid under structural stability and non-characteristic interface assumptions.

Error estimates depend on relaxation time and boundary layer effects.

The DG scheme is proven to be L2 stable and effective in numerical tests.

Abstract

This work is concerned with coupling conditions for linear hyperbolic relaxation systems with multiple relaxation times. In the region with small relaxation time, an equilibrium system can be used for computational efficiency. Under the assumption that the relaxation system satisfies the structural stability condition and the interface is non-characteristic, we derive a coupling condition at the interface to couple the two systems in a domain decomposition setting. We prove the validity by the energy estimate and Laplace transform, which shows how the error of the domain decomposition method depends on the smaller relaxation time and the boundary layer effects. In addition, we propose a discontinuous Galerkin (DG) scheme for solving the interface problem with the derived coupling condition and prove the L2 stability. We validate our analysis on the linearized Carleman model and the linearized Grad's moment system and show the effectiveness of the DG scheme.