Application of Block Krylov Subspace Spectral Methods to Maxwell’s Equations

Abstract

Ever since its introduction by Kane Yee over forty years ago, the finite‐difference time‐domain (FDTD) method has been a widely‐used technique for solving the time‐dependent Maxwell’s equations. This paper presents an alternative approach to these equations in the case of spatially‐varying electric permittivity and/or magnetic permeability, based on Krylov subspace spectral (KSS) methods. These methods have previously been applied to the variable‐coefficient heat equation and wave equation, and have demonstrated high‐order accuracy, as well as stability characteristic of implicit time‐stepping schemes, even though KSS methods are explicit. KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Gene Golub and Gerard Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. We show how they can be generalized to coupled systems of equations, such as Maxwell’s equations, by choosing appropriate basis functions that, while induced by this coupling, still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields. We also discuss the implementation of appropriate boundary conditions for simulation on infinite computational domains, and how discontinuous coefficients can be handled.