An approximation scheme for an Eikonal Equation with discontinuous coefficient

TL;DR

This paper introduces a semi-Lagrangian numerical scheme for solving a class of stationary Hamilton-Jacobi equations with discontinuous coefficients, providing error estimates and applications in control and image processing.

Contribution

It proposes a new semi-Lagrangian scheme tailored for Hamilton-Jacobi equations with discontinuous coefficients and establishes its error bounds.

Findings

The scheme accurately approximates viscosity solutions with discontinuities.

Error estimates are proven in an integral norm.

Applications demonstrated in control and image processing tasks.

Abstract

We consider the stationary Hamilton-Jacobi equation where the dynamics can vanish at some points, the cost function is strictly positive and is allowed to be discontinuous. More precisely, we consider special class of discontinuities for which the notion of viscosity solution is well-suited. We propose a semi-Lagrangian scheme for the numerical approximation of the viscosity solution in the sense of Ishii and we study its properties. We also prove an a-priori error estimate for the scheme in an integral norm. The last section contains some applications to control and image processing problems.