Numerical Analysis of AVF Methods for Three-Dimensional Time-Domain Maxwell’s Equations

Abstract

We propose two schemes [AVF(2) and AVF(4)] for Maxwell's equations, by discretizing the Hamiltonian formulation with Fourier pseudospectral method for spatial discretization and average vector field method for time integration. Both AVF(2) and AVF(4) hold the two Hamiltonian energies automatically, while being energy-, momentum-and divergence-preserving, unconditionally stable, non-dissipative and spectral accurate. Rigorous error estimates are obtained for the proposed schemes. The numerical dispersion relations are also investigated. Numerical experiments support well the theoretical analysis results. The proposed schemes are valid for the regular domain, but invalid for the domain with complex geometries.

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“…However, few convergent results of the Fourier pseudo-spectral method are available. Recently, Cai et al [13][14][15] developed and analyzed different structurepreserving Fourier pseudo-spectral methods for the 3D Maxwell's equation. But unconditionally convergent results on Fourier pseudo-spectral method for nonlinear PDEs have not been obtained.…”

An Energy-Preserving Scheme for the Coupled Gross-Pitaevskii Equations

“…In this paper, we intend to propose an energy-preserving scheme for the CGP equations by the average vector field (AVF) method which was introduced by Quispel et al recently [24,28]. One of the excellent advantages of the AVF method is that it can preserve the energy invariant automatically [7,8,13]. Moreover, it is relatively easy to construct temporal high order energy-preserving schemes for Hamiltonian systems.…”

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