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

Abstract

This work is concerned with coupling conditions for linear hyperbolic relaxation systems with multiple relaxation times. In the region with a small relaxation time, an equilibrium system can be used for computational efficiency. The key assumption is that the relaxation system satisfies Yong’s structural stability condition [J. Differential Equations, 155 (1999), pp. 89–132]. For the non-characteristic case, we derive a coupling condition at the interface to couple 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) numerical scheme for solving the interface problem with the derived coupling condition and prove the L 2 L^2 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.