Volume 15, Issue 6
Construction and Analysis of Weighted Sequential Splitting FDTD Methods for the 3D Maxwell's Equations

Vrushali A. Bokil and Puttha Sakkaplangkul

Int. J. Numer. Anal. Mod., 15 (2018), pp. 747-784.

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  • 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 6= 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 6= 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.

  • History

Published online: 2018-08

  • AMS Subject Headings

65M06, 65M12, 65Z05.

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