Discrete Shocks for Finite Difference Approximations to Scalar Conservation Laws

Numerical simulations often provide strong evidences for the existence and stability of discrete shocks for certain finite difference 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 flux function of the scheme is assumed to be twice differentiable but the scheme can be nonlinear and of any order of accuracy. To prove existence and stability, we show that it would suffice to verify some simple inequalities, which can usually be done using computers. As examples, we use the framework to give an unified proof of the existence of continuous discrete shock profiles for a modified first-order Lax--Friedrichs scheme and the second-order Lax--Wendroff scheme. We also show the existence and stability of discrete shocks for a third-order weighted essentially nonoscillatory (ENO) scheme.