Detailed analysis of the effects of stencil spatial variations with arbitrary high-order finite-difference Maxwell solver

TL;DR

This paper develops an analytical method to evaluate discretization errors in high-order finite-difference Maxwell solvers, accounting for boundary modifications like PMLs and domain decomposition, ensuring accurate electromagnetic simulations.

Contribution

The paper introduces a general analytical approach for assessing errors in high-order Maxwell solvers with arbitrary stencil modifications, validated against numerical simulations.

Findings

Analytical predictions match simulation results accurately.

High-order Maxwell solvers can be combined with domain decomposition with minimal error.

The approach works for PMLs and pseudo-spectral limits.

Abstract

Due to discretization effects and truncation to finite domains, many electromagnetic simulations present non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the result. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of the errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical discretization errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solver and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.