Discrete Shocks for Finite Diierence Approximations to Scalar Conservation Laws 1

Numerical simulations often provide strong evidences for the existence and stability of discrete shocks for certain nite diierence schemes approximating conservation laws. This paper presents a framework for converting such numerical observations to mathematical proofs. The framework is applicable to conservative schemes approximating stationary shocks of one dimensional scalar conservation laws. The numerical ux function of the scheme is assumed to be twice diierentiable but the sheme can be nonlinear and of any order of accuracy. To prove existence and stability, we show that it would be suuce to verify some simple inequalities, which can usually be done using computers. As examples, we use the framework to give an uniied proof of the existence of continuous discrete shock prooles for a modiied rst order Lax-Friedrichs scheme and the second order Lax-Wendroo scheme. We also show the existence and stability of discrete shocks for a third order weighted ENO scheme.