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

中文翻译：

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可对称线性双曲系统有限体积离散化的长时间行为

本文致力于研究一般可对称线性双曲系统在有界域上的有限体积逼近的长时间行为。在连续情况下这个问题是非常困难的，如果不引入耗散，$\omega$-limit 集（即所有可能的长时间限制的集合）可能很大且难以描述。在本文中，我们证明了一般情况下，对于稳定的有限体积方案，当时间趋于无穷时，离散解会收敛到稳态。该属性是用于稳定离散化的数值耗散机制的直接结果。我们应用这个结果来确定波浪系统的几个稳定的长时限，并与非线性欧拉系统的低马赫数问题进行正式联系。