Energy/dissipation-preserving Birkhoffian multi-symplectic methods for Maxwell's equations with dissipation terms

Abstract

In this paper, we propose two new energy/dissipation-preserving Birkhoffian multi-symplectic methods (Birkhoffian and Birkhoffian box) for Maxwell's equations with dissipation terms. After investigating the non-autonomous and autonomous Birkhoffian formalism for Maxwell's equations with dissipation terms, we first apply a novel generating functional theory to the non-autonomous Birkhoffian formalism to propose our Birkhoffian scheme, and then implement a central box method to the autonomous Birkhoffian formalism to derive the Birkhoffian box scheme. We have obtained four formal local conservation laws and three formal energy global conservation laws. We have also proved that both of our derived schemes preserve the discrete version of the global/local conservation laws. Furthermore, the stability, dissipation and dispersion relations are also investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, dissipation-preserving for Maxwell's equations in a perfectly matched layer (PML) medium and have second order accuracy in both time and space. Numerical experiments for problems with exact theoretical results are given to demonstrate that the Birkhoffian multi-symplectic schemes are much more accurate in preserving energy than both the exponential finite-difference time-domain (FDTD) method and traditional Hamiltonian scheme. We also solve the electromagnetic pulse (EMP) propagation problem and the numerical results show that the Birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation.

A Conformal Energy-Conserved Method for Maxwell's Equations with Perfectly Matched Layers

“…Lemma 2.1. [32] Let H n and E n be the solutions of (2.3) and satisfy the periodic boundary condition, then the following conformal energy conservation laws hold 3) is perhaps not always met in practical computations, but it is the key condition of the PML for solving unbounded electromagnetic problems. Therefore, the investigation of numerical methods for the Maxwell's equations in such special case is also very important.…”

“…Firstly, we will focus on the rate of convergence and the conformal energy conservation laws of the 2D Maxwell's equations with the PML. Furthermore, some comparisons between the conformal Fourier pseudo-spectral scheme (CFPS) and the Birkhoffian multisymplectic scheme (BMSS) [32] We set the analytic solutions (4.9)-(4.11) at t=0 on the domain Ω=[0,1]×[0,1] as initial conditions. Table 1 shows the temporal errors and the convergence rates of the different schemes withσ = 0.1 and N x = N y = 1024 at T = 1.…”

“…Since Er 4 y and Er 5 y are close to Er 4 x and Er 5 x , respectively, the values of Er 4 y and Er 5 y are not displayed. Moreover, it has shown that both the popular exponential FDTD Yee's scheme [2] and the traditional multi-symplectic scheme can only approximately preserve the conformal energy conservation laws [32], which implies that the CFP scheme provides distinct advantages in preserving the conformal conservation laws than the exponential FDTD Yee's scheme and the multi-symplectic scheme.…”

“…In Ref. [32], Birkhoffian multi-symplectic methods for the Maxwell's equations with the PML were investigated by Su and Li. However, most of existing methods have low order accuracy in space and the rigorous error estimate is not established well.…”

Ergodic numerical approximations for stochastic Maxwell equations

“…The damping terms σE and σH may be induced by conductivity of the medium or by the perfectly matched layer technique (see e.g. [15,19]). It is assumed that σ ∈ W 1,∞ (D) and σ ≥ σ 0 > 0 for a constant σ 0 .…”