Error Analysis of a Second-Order Locally Implicit Method for Linear Maxwell's Equations

Abstract

Abstract.In this paper we consider the full discretization of linear Maxwell's equations on spatial grids which are locally refined. For such problems, explicit time integration schemes become ine cient because the smallest mesh width results in a strict CFL condition. Recently locally implicit time integration methods have become popular in overcoming the problem of so called grid-induced sti↵ness. Various such schemes have been proposed in the literature and have been shown to be very e cient. However, a rigorous analysis of such methods is still missing. In fact, the available literature focuses on error bounds which are valid on a fixed spatial mesh only but deteriorate in the limit where the smallest spatial mesh size tends to zero. Moreover, some important questions cannot be answered without such an analysis. For example, it has not yet been studied which elements of the spatial mesh enter the CFL condition.In this paper we provide such a rigorous analysis for a locally implicit scheme proposed by Verwer [15] based on a variational formulation and energy techniques.

Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations

“…By the arguments of the proof of Lemma 2.2 in [23], we obtain that the discrete curl operator satisfies the discrete version of Green's formula (2.2),…”

“…Like Abboud et al [1] (and later also [5]) for the acoustic wave equation, we start from a symmetrized weak first-order formulation of Maxwell's equations. In the interior this is discretized by a discontinuous Galerkin (dG) method in space [14,21,23] together with the explicit leapfrog scheme in time [19]. The boundary integral terms are discretized by standard boundary element methods in space and by convolution quadrature (CQ) in time [25,26].…”

“…In the interior these coefficients may be space-dependent and discontinuous. In the latter case the equations can be discretized in space with the dG method as described in [23].…”

Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation

“…Despite the many different explicit LTS methods that were proposed and successfully used for wave propagation in recent years, a rigorous fully discrete space-time convergence theory is still lacking. In fact, convergence has been proved only for the method of Collino et al [12,11,32] and very recently for the locally implicit method for Maxwell's equations by Verwer [47,17,30], neither fully explicit. Indeed, the difficulty in proving convergence of fully explicit LTS methods is twofold.…”

“…from(35) and the second equation in(29) from(36)we obtain e (n+1/2) v,S,∆t = e (n−1/2) v,S,∆t − (∆t) A S,p e Eliminating the term e (n+1/2) v,S,∆t in the second equation by using the first one yields e (n+1/2) v,S,∆t = e (n−1/2) v,S,∆t − (∆t) A S,p e (n) u,S,∆t + (∆t) ∆ (n+1/2) 1 , e (n+1) u,S,∆t = (∆t) e (n−1/2) v,S,∆t + e (n) u,S,∆t − (∆t) 2 A S,p eWe rewrite it in operator form by using the operator S as in(30) …”