Discrete shocks for difference approximations to systems of conservation laws (Q1064029)

The existence of planar weak discrete shocks for a wide class of difference approximations to systems of conservation laws is proved. The difference schemes have to be conservative, k-th order accurate, and, roughly speaking, \((k+1)-t\) 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 \(y^{(k)}=y^ 2-1\) for \(k=3\) intersect transversally.