ASpectral Time-DomainMethod for Computational Electrodynamics

Ever since its introduction by Kane Yee over forty years ago, the finitedifference time-domain (FDTD) method has been a widely-used technique for solving the time-dependent Maxwell’s equations that has also inspired many other methods. 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 andwave equation, and have demonstrated high-order accuracy, as well as stability characteristic of implicit timestepping schemes, even though KSS methods are explicit. KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed byGolub andMeurant for approximating elements of functions ofmatrices 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 application of block KSS methods to problems involving non-self-adjoint spatial differential operators, which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices. AMS subject classifications: 65M10, 78A48