A Krylov‐subspace based solver for the linear and nonlinear Maxwell equations

Abstract

We describe an efficient Krylov-subspace based operator-exponential approach for solving the Maxwell equations. This solver exhibits excellent stability properties and high-order time-stepping capabilities that allow to address nonlinear wave propagation phenomena and/or coupled system dynamics. Furthermore, the usage of a non-uniform spatial grid facilitates the realization of a high-order spatial discretization in the presence of discontinuous material properties. This ideally complements the time-stepping capabilities of our solver so that complex nano-photonic problems may be treated with high accuracy and efficiency. We illustrate these features through an analysis of the prototypical problem of spontaneous emission from a collection of two-level atoms that are embedded in finite Photonic Crystals and are exposed to dephasing as well as non-radiative decay processes.

Discontinuous Galerkin time-domain computations of metallic nanostructures

“…[27]. However, since within DGTD the time-stepping is essentially disentangled from the spatial discretization (as opposed to the situation in FDTD), more sophisticated methods such as Krylov-subspace based operator exponential techniques [28] could provide an even more efficient solver.…”

Numerical solutions of the time‐dependent Schrödinger equation with position‐dependent effective mass

“…The basic idea of the Krylov subspace approach is to project the exponential of a large matrix/operator onto a relatively small-sized Krylov subspace where calculating the exponential is significantly less computationally expensive [63]. The Krylov subspace method-based exponential integration has been applied successfully for solving many different problems [13,22,32,35,69], especially in differential equations, such as Maxwell's equations in time [13,15,56], large system of differential equations [35], multifrequency optical response [14], reactor kinetics equation [4], fast pricing of options equations [71], fluid dynamics equations [64], shallow water equations [30], Dirac equation [9], incompressible Navier-Stokes equations [21], etc. We shall apply it for solving the subproblem of Equation ( 1) that is related to the kinetic operator as well.…”

POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations

“…MOR method is useful for accelerating simulations in many fields of science and engineering [16,17,18,19,20,21,22]. In particular, MOR method is also widely used in the context of electromagnetics [19,23,24,25,26,27,28,29,30,31,32]. The overall goal of MOR can be stated as to reduce the computational requirements while maintaining an acceptable level of accuracy.…”