Dispersion Analysis of Finite Difference and Discontinuous Galerkin Schemes for Maxwell's Equations in Linear Lorentz Media

TL;DR

This paper analyzes the numerical dispersion errors of finite difference and discontinuous Galerkin schemes for Maxwell's equations in Lorentz media, providing guidance for optimal discretization choices based on detailed dispersion analysis.

Contribution

It offers a comprehensive dispersion analysis of FD and DG schemes coupled with semi-implicit time integrators for Maxwell's equations in dispersive media, highlighting their advantages and limitations.

Findings

Dispersion errors depend on spatial and temporal discretizations and model parameters.

High order schemes can reduce dispersion errors over certain frequency ranges.

Limitations of second order temporal schemes are identified.

Abstract

In this paper, we consider Maxwell's equations in linear dispersive media described by a single-pole Lorentz model for electronic polarization. We study two classes of commonly used spatial discretizations: finite difference methods (FD) with arbitrary even order accuracy in space and high spatial order discontinuous Galerkin (DG) finite element methods. Both types of spatial discretizations are coupled with second order semi-implicit leap-frog and implicit trapezoidal temporal schemes studied in our previous research [5,6]. By performing detailed dispersion analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of dispersion error can be made taking into account the model parameters, and mesh sizes in the design of the two types of schemes. The results for the numerical dispersion analysis can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models. The numerical dispersion analysis of the fully discrete FD and DG schemes, for the dispersive Maxwell model considered in this paper, clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations, their order of accuracy, mesh discretization parameters and model parameters. The results obtained here cannot be arrived at by considering discretizations of Maxwell's equations in free space. In particular, our results contrast the advantages and disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors. Finally, we highlight the limitations of the second order accurate temporal discretizations considered.