On backward Euler approximations for systems of conservation laws

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: (1) a scalar conservation law, where the jump \(u_l-u_r\) can be arbitrarily large, and (2) a strictly hyperbolic system, assuming that the jump \(u_l-u_r\) occurs in a genuinely nonlinear family and is sufficiently small.