Nonlinear stability of discrete shocks for systems of conservation laws (Q1314295)

The authors study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs difference scheme for systems of nonlinear hyperbolic conservation laws. The main result states that a single weak discrete shock is stable in \(L^ p\) for all \(p \geq 1\) if the initial perturbations sum to zero. Further, it is shown that if the solution of the Riemann problem for the far-field states consists only of shocks, one from each characteristic family, then the corresponding discrete solution is stable in \(L^ p\) for \(p \geq 2\). The methods used include a weighted estimate and the characteristic energy method, based on the essential monotonicity of the Lax-Friedrichs scheme.