Godunov scheme for Maxwell's equations with Kerr nonlinearity

TL;DR

This paper develops and analyzes a Godunov scheme for solving Maxwell's equations with Kerr nonlinearity, demonstrating its effectiveness through implementation and comparison with relaxation approximations.

Contribution

It constructs a Riemann solver for the full 6x6 Kerr Maxwell system and implements the scheme in multiple dimensions, advancing numerical methods for nonlinear optics.

Findings

Solutions exist for all data in the Riemann problem.

Numerical results closely match Kerr-Debye relaxation approximations.

The scheme effectively captures nonlinear optical phenomena.

Abstract

We study the Godunov scheme for a nonlinear Maxwell model arising in nonlinear optics, the Kerr model. This is a hyperbolic system of conservation laws with some eigenvalues of variable multiplicity, neither genuinely nonlinear nor linearly degenerate. The solution of the Riemann problem for the full-vector 6x6 system is constructed and proved to exist for all data. This solution is compared to the one of the reduced Transverse Magnetic model. The scheme is implemented in one and two space dimensions. The results are very close to the ones obtained with a Kerr-Debye relaxation approximation.