  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.
