Construction and Analysis of Weighted Sequential Splitting FDTD Methods for the 3D Maxwell's Equations
Year: 2018
Author: Vrushali A. Bokil, Puttha Sakkaplangkul
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 6 : pp. 747–784
Abstract
In this paper, we present a one parameter family of fully discrete Weighted Sequential Splitting (WSS)-finite difference time-domain (FDTD) methods for Maxwell’s equations in three dimensions. In one time step, the Maxwell WSS-FDTD schemes consist of two substages each involving the solution of several 1D discrete Maxwell systems. At the end of a time step we take a weighted average of solutions of the substages with a weight parameter $θ$, $0 ≤ θ ≤ 1$. Similar to the Yee-FDTD method, the Maxwell WSS-FDTD schemes stagger the electric and magnetic fields in space in the discrete mesh. However, the Crank-Nicolson method is used for the time discretization of all 1D Maxwell systems in our splitting schemes. We prove that for all values of $θ$, the Maxwell WSS-FDTD schemes are unconditionally stable, and the order of accuracy is of first order in time when $θ\neq 0.5$, and of second order when $θ = 0.5$. The Maxwell WSS-FDTD schemes are of second order accuracy in space for all values of $θ$. We prove the convergence of the Maxwell WSS-FDTD methods for all values of the weight parameter $θ$ and provide error estimates. We also analyze the discrete divergence of solutions to the Maxwell WSS-FDTD schemes for all values of $θ$ and prove that for $θ\neq 0.5$ the discrete divergence of electric and magnetic field solutions is approximated to first order, while for $θ = 0.5$ we obtain a third order approximation to the exact divergence. Numerical experiments and examples are given that illustrate our theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2018-IJNAM-12608
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 6 : pp. 747–784
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 38
Keywords: Maxwell’s equations Yee scheme Crank-Nicolson method operator splitting weighted sequential splitting.