Krylov subspace exponential time domain solution of Maxwell’s equations in photonic crystal modeling

Abstract

MSC: 65F60 65F30 65N22 65L05 35Q61 Keywords: Finite difference time domain (FDTD) method Matrix exponential Maxwell's equations Krylov subspace methods Photonic crystals a b s t r a c tThe exponential time integration, i.e., time integration which involves the matrix exponential, is an attractive tool for time domain modeling involving Maxwell's equations. However, its application in practice often requires a substantial knowledge of numerical linear algebra algorithms, such as Krylov subspace methods.In this note we discuss exponential Krylov subspace time integration methods and provide a simple guide on how to use these methods in practice. While specifically aiming at nanophotonics applications, we intentionally keep the presentation as general as possible and consider full vector Maxwell's equations with damping (i.e., with nonzero conductivity terms).Efficient techniques such as the Krylov shift-and-invert method and residual-based stopping criteria are discussed in detail. Numerical experiments are presented to demonstrate the efficiency of the discussed methods and their mesh independent convergence. Some of the algorithms described here are available as Octave/Matlab codes from www.math.utwente.nl/~botchevma/expm/.

Exponential Krylov time integration for modeling multi-frequency optical response with monochromatic sources

“…For the SaI method, the relations above hold with A replaced by (I +γA) −1 . The SaI transformation results in a better approximation in the Krylov subspace of the eigenmodes corresponding to the small in modulus eigenvalues [38,50] and, hence, is favorable for solving the time dependent problem (7). Indeed, for some classes of the discretized differential operators A, such as the discretizations of parabolic PDEs with a numerical range along the positive real axis, one can show that the convergence of the SaI methods is mesh independent [50,25].…”

“…Indeed, for some classes of the discretized differential operators A, such as the discretizations of parabolic PDEs with a numerical range along the positive real axis, one can show that the convergence of the SaI methods is mesh independent [50,25]. Although these results can not be extended to wave-type equations in a straightforward manner, a mesh independent convergence is observed in practice for the Maxwell equations with damping in [7]. For the SaI Krylov method we define the matrix H m as the inverse shift-and-invert transformation:…”

“…To numerically solve the Maxwell equations, we first make the equations dimensionless. We follow the standard dimensionless procedure as described, e.g., in [7]. In the two-dimensional computational model, the unknown field components are H x (x, y), H y (x, y) (magnetic field) and E z (x, y) (electric field) as the incident light originates from a point electric dipole.…”

“…We choose the parameter γ in the SaI Krylov subspace method by making cheap trial runs on a coarse mesh (in this case with resolution 16 points per unit length) and using the chosen γ for all the meshes [7]. This is possible because the SaI methods usually exhibit a mesh independent convergence [50,25,7]. We also look at the number and size of the spurious eigenvalues in the back SaI transformed matrix H m .…”

A nested Schur complement solver with mesh-independent convergence for the time domain photonics modeling

“…Unlike the ADI preconditioner, the FS preconditioner is not restricted to the finite difference approximations on Cartesian meshes. The linear system (5), with either the FS or ADI preconditioner applied from the right, can be written as…”