A high-order accurate scheme for Maxwell's equations with a generalized dispersive material model

Abstract

A high-order accurate scheme for solving the time-domain Maxwell's equations with a generalized dispersive material model is described. The equations for the electric field are solved in second-order form, and a general dispersion model is treated with the addition of one or more polarization vectors which obey a set of auxiliary differential equations (ADE). Numerical methods are developed for both second-order and fourth-order accuracy in space and time. The equations are discretized using finite-differences, and advanced in time with a single-stage, three-level, space–time scheme which remains stable up to the usual explicit CFL restriction, as proven using mode analysis. Because the equations are treated in their second-order form, there is no need for grid staggering, and instead a collocated grid is used. Composite overlapping grids are used to treat complex geometries with boundary-conforming grids, and a high-order upwind dissipation is added to ensure robust and stable approximations on overlapping grids. Numerical results in two and three space dimensions confirm the accuracy and stability of the new schemes.