Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities

Abstract

We consider the Hamilton-Jacobi equation of eikonal type

where $H$ is convex and $f$ is allowed to be discontinuous. Under a suitable assumption on $f$ we prove a comparison principle for viscosity sub- and supersolutions in the sense of Ishii. Furthermore, we develop an error analysis for a class of finite difference schemes, which are monotone, consistent and satisfy a suitable stability condition.