Discrete shocks for difference approximations to systems of conservation laws

The existence of weak discrete shocks for a wide class of difference approximations to systems of conservation laws is proved. The difference schemes have to be conservative, kth order accurate, and, roughly speaking, (k + 1)th order dissipative, where k = 1 or 3. The proof makes use of the central manifold theorem for an implicit map and of the fact that the stable and unstable manifolds of the differential flow y^(k) = y² − 1 for K = 3 intersect transversally.