The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations

Abstract

For the time integration of edge finite element discretizations of the three-dimensional Maxwell equations, we consider the Gautschi cosine scheme where the action of the matrix function is approximated by a Krylov subspace method. First, for the spacediscretized edge finite element Maxwell equations, the dispersion error of this scheme is analyzed in detail and compared to that of two conventional schemes. Second, we show that the scheme can be implemented in such a way that a higher accuracy can be achieved within less computational time (as compared to other implicit schemes). We also analyzed the error made in the Krylov subspace matrix function evaluations. Although the new scheme is unconditionally stable, it is explicit in structure: as an explicit scheme, it requires only the solution of linear systems with the mass matrix.

Efficient time integration for discontinuous Galerkin approximations of linear wave equations

“…Neither a further reduction of the time step nor using a higher order method improve the overall accuracy. The regularity in space is limited by the regularity of the initial function u 0 ∈ H 2 (Ω) 3 \ H 3 (Ω) 3 , so that we observe a convergence factor between 4 and 8 for quadratic elements in space.…”

“…It is well known that polynomial Krylov subspace methods for the matrix exponential exp(τ A)v always converge superlinearly but unfortunately, for the wave equation, the onset of superlinear convergence behavior only starts after τ A steps [21]. Nevertheless, these methods have been successfully applied to wave problems, see, e.g., [3,7,30,41]. In contrast, it has been shown recently in [12,14,16] that rational Krylov subspace methods converge independently of τ A and even for unbounded operators A.…”

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

“…In our limited experience, exponential time integrators are computationally efficient for Maxwell's equations primarily in the first setting. For instance, the CO2 scheme appears to be much more efficient than EK2 in the experiments from [41], even though some promising, in terms of computational efficiency, results with exponential integration are reported in [59]. An approach to reduce the number of matrix function actions and, hence, to increase efficiency of the exponential integration is presented in [17].…”

A block Krylov subspace time‐exact solution method for linear ordinary differential equation systems

“…The first two sources of the error are by far dominant, but can be easily controlled when 560 M. A. BOTCHEV the approximation is constructed (see [27] and Section 4), thus giving possibility for an adaptive approximation procedure. With (6), the original initial-value problem (2) takes the form…”

“…These are time integration schemes involving the matrix exponential and related matrix functions. An attractive feature of exponential time integrators is a combination of excellent stability and accuracy properties, with the latter being usually better than in the standard implicit time integrators [3][4][5][6]. The interest in exponential time integration is due to the new, challenging applications [7-9] as well as to the recent progress in Krylov subspace techniques to compute actions of matrix functions for large matrices (e.g., [10][11][12][13][14][15][16][17][18][19][20]).…”