Existence of semi-discrete shocks

The approximation of shock waves by finite difference schemes is considered. This question has been investigated by many authors, but mainly under some restrictions on discrete wave speeds. Basic works are due to Majda and Ralston (rational speed) and, more recently, to Liu and Yu (Diophantine speed). The main purpose of the present work is to obtain shock profiles of arbitrary speed for rather general schemes. As a first step, we deal with semi-discretizations in space. For dissipative and non-resonant schemes, using the terminology of Majda and Ralston, we show the existence of shock profiles of small strength. For this we prove a center manifold theorem for a functional differential equation of mixed type (with both delay and advance). An additional invariance principle enables us to find semi-discrete shocks as heteroclinic orbits on the center manifold exhibited. This result generalizes a previous one of the first author, dealing with the special "upwind" scheme. In particular, it holds for the Godunov scheme and for a semi-discrete version of the Lax-Friedrichs scheme also known as the Rusanov scheme.