Analysis of a Backward Euler-type Scheme for Maxwell's Equations in a Havriliak-Negami Dispersive Medium

TL;DR

This paper develops and analyzes a stable backward Euler-type numerical scheme for Maxwell's equations in Havriliak-Negami dispersive media, ensuring energy decay, efficiency, and accuracy, with applications to permittivity recovery.

Contribution

It introduces a novel unconditionally stable semi-discrete scheme with a fast convolution algorithm for H-N media and demonstrates its effectiveness through numerical experiments.

Findings

The scheme is unconditionally stable with monotonic energy decay.

The fast convolution algorithm reduces computational burden.

Numerical results confirm high accuracy and efficiency.

Abstract

For the Maxwell's equations in a Havriliak-Negami (H-N) dispersive medium, the associated energy dissipation law has not been settled at both continuous level and discrete level. In this paper, we rigorously show that the energy of the H-N model can be bounded by the initial energy and the model is well-posed. We analyse a backward Euler-type semi-discrete scheme, and prove that the modified discrete energy decays monotonically in time. Such a strong stability ensures that the scheme is unconditionally stable. We also introduce a fast temporal convolution algorithm to alleviate the burden of the history dependence in the polarisation relation involving the singular kernel with the Mittag-Leffler function with three parameters. We provide ample numerical results to demonstrate the efficiency and accuracy of a full-discrete scheme via a spectra-Galerkin method in two dimensions. Finally, we consider an interesting application in the recovery of complex relative permittivity and some related physical quantities.