An accurate and stable fourth order finite difference time domain method

A long-stencil fourth order finite difference method over a Yee-grid is developed to solve Maxwell’s equations. The different variables are located at staggered mesh points, and a symmetric image formula is introduced near the boundary. The introduction of these symmetric ghost grid points assures the stability of the boundary extrapolation, and in turn a complete set of purely imaginary eigenvalues are given for the fourth-order discrete curl operators for both electric and magnetic fields. Subsequently, the four-stage Jameson method integrator constrained by a pre-determined time step is utilized to produce a stable full fourth order accuracy in both time and space. The accuracy of the developed numerical scheme has been validated by comparing its results to the closed form solutions for a rectangular cavity.

I. Introduction

The classical Finite Difference Time Domain (FDTD) approach introduced by Yee has been widely used in the solution of problems in electromagnetics [1]. The scheme proved to be second order accurate in space and time. However, as always in the case of a second order method, the Yee scheme begins to accumulate phase errors as time grows large, especially for electrically large domains or for late-time analysis. Because of this, several attempts have been made to extend this scheme to have a fourth order accuracy [2]. Fourth order methods allow a larger time step and courser spatial mesh to be used while maintaining the same accuracy (effectively increasing the simulation speed), or can increase the accuracy for a given mesh spacing and time step.