On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations

TL;DR

This paper develops a semi-Lagrangian numerical scheme for approximating solutions to a Hamilton-Jacobi-transport system in geometrical optics, proving its well-posedness and convergence to the viscosity-measure solution.

Contribution

It introduces a new semi-Lagrangian scheme for the Hamilton-Jacobi-transport system and proves its convergence and well-posedness.

Findings

The scheme is well-posed.

The scheme converges to the viscosity-measure solution.

The method is applicable to geometrical optics problems.

Abstract

In the present article, we study the numerical approximation of a system of Hamilton-Jacobi and transport equations arising in geometrical optics. We consider a semi-Lagrangian scheme. We prove the well posedness of the discrete problem and the convergence of the approximated solution toward the viscosity-measure valued solution of the exact problem.