A Spectral Time-Domain Method for Computational Electrodynamics (Advances in Applied Mathematics and Mechanics, 2021)
Ever since its introduction by Kane Yee over forty years ago, the finite-difference time-domain (FDTD) method has been awidely-used technique for solving the time-dependent Maxwell'sequations that has also inspired many other methods. This paperpresents an alternative approach to these equations in the case ofspatially-varying electric permittivity and/or magneticpermeability, based on Krylov subspace spectral (KSS) methods. Thesemethods have previously been applied to the variable-coefficientheat equation and wave equation, and have demonstrated high-orderaccuracy, as well as stability characteristic of implicittime-stepping schemes, even though KSS methods are explicit.  KSSmethods for scalar equations compute each Fourier coefficient of thesolution using techniques developed by Golub and Meurant forapproximating elements of functions of matrices by Gaussianquadrature in the spectral, rather than physical, domain. We showhow they can be generalized to coupled systems of equations, such asMaxwell's equations, by choosing appropriate basis functions that,while induced by this coupling, still allow efficient and robustcomputation of the Fourier coefficients of each spatial component ofthe electric and magnetic fields.  We also discuss the applicationof block KSS methods to problems involving non-self-adjoint spatialdifferential operators, which requires a generalization of the blockLanczos algorithm of Golub and Underwood to unsymmetric matrices.