*Submitted Paper*

**Inserted:** 28 nov 2021

**Last Updated:** 28 nov 2021

**Year:** 2021

**Abstract:**

We study approximate solutions to a hyperbolic system of conservation laws, constructed by a backward Euler scheme, where time is discretized while space is still described by a continuous variable $x\in \mathbb{R}$. We prove the global existence and uniqueness of these approximate solutions, and the invariance of suitable subdomains. Furthermore, given a left and a right state $u_l, u_r$ connected by an entropy-admissible shock, we construct a traveling wave profile for the backward Euler scheme connecting these two asymptotic states in two main cases. Namely: (i) a scalar conservation law, where the jump $u_l-u_r$ can be arbitrarily large, and (ii) a strictly hyperbolic system, assuming that the jump $u_l-u_r$ occurs in a genuinely nonlinear family and is sufficiently small.

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