Communications in Mathematical Sciences

Volume 13 (2015)

Number 8

Godunov scheme for Maxwell’s equations with Kerr nonlinearity

Pages: 2195 – 2222

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n8.a10

Author

Denise Aregba-Driollet (Institut de Mathématiques de Bordeaux, Talence, France)

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, that are neither genuinely nonlinear nor linearly degenerate. The solution of the Riemann problem for the full-vector $6 \times 6$ system is constructed and proven 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 by a Kerr–Debye relaxation approximation.

Keywords

Godunov, Riemann problem, finite volumes, relaxation, Kerr model, Kerr–Debye model

2010 Mathematics Subject Classification

35L65, 35L67, 65M08, 78-04

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Published 3 September 2015