Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systems

Abstract This article is dedicated to the long time behavior of a finite volume approximation general symmetrizable linear hyperbolic system on bounded domain. In continuous case this problem very difficult, and $\omega $–limit set (namely all possible limits) may be large complicated depict if no dissipation introduced. we prove that in general, with stable scheme, discrete solution converges steady state when goes infinity. property direct consequence numerical mechanisms used for stabilizing discretization. We apply result determining limit several stabilizations wave system, perform formal link low Mach number nonlinear Euler system. Numerical experiments are performed confirming theoretical results obtained.