Convergence of an ADI splitting for Maxwell’s equations

Abstract

The convergence of an alternating direction implicit method for Maxwell's equations on product domains is investigated. Unlike the classical Yee scheme and most other integrators proposed in the literature, this method is both unconditionally stable and computationally cheap. We prove second-order convergence of the time-discretization in the framework of operator semigroup theory. In contrast to formal considerations based on Taylor expansions, our convergence analysis respects the unboundedness of the involved differential operators. The proofs are based on results concerning the regularity of the Cauchy problems, which then allow to apply an abstract convergence proof by Hansen and Ostermann [13].

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

“…Recently an efficient alternating direction implicit (ADI) preconditioner is proposed and analyzed for solving time-dependent Maxwell equations [17] discretized in space by finite differences. In [8] de Cloet, Marissen and Westendorp compared performance of this ADI preconditioner with another preconditioner based on the field splitting (i.e., a splitting into the magnetic and electric fields).…”

Error Analysis of an Energy Preserving ADI Splitting Scheme for the Maxwell Equations

“…Since we can only guarantee that the solution belong to H 2 , we thus have to measure the error in H −1 . For the non-conservative ADI system from [20,24], the error formulas of [8,9,10,14] only contain double products of A and B. In these papers we were thus able to establish second convergence in L 2 , assuming one more degree of initial regularity.…”

“…The first ADI method for the Maxwell equations (without currents, conductivity and charges) has been proposed independently in [20] and [24]. The convergence of the semidiscretization with this ADI method has been analyzed in [14] on R 3 and on a cuboid Q with perfectly conducting boundary conditions. For the Maxwell system with sources, currents and conductivity, second order convergence in a weak sense has been shown in [9].…”

“…It is well-known that the solution of the Maxwell equations has constant energy in absence of sources, currents and conductivity. The ADI method considered in [8,9,10,14,20,24], however, is based on the Peaceman-Rachford scheme and hence does not conserve the energy of the numerical solution. Unconditionally stable ADI schemes which do preserve energy have been proposed and investigated, e.g., in [4,5,12,13,17,18].…”

Exponential integrators for stochastic Maxwell's equations driven by Itô noise

“…In addition, the operator A generates a unitary C 0 -group S(t) := exp(tA) via Stone's theorem, see for example [17]. According to the definition of unitary groups, one has…”