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

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

Published online: 2018-08

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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 $θ\neq 0.5$, 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.

• Keywords

Maxwell’s equations, Yee scheme, Crank-Nicolson method, operator splitting, weighted sequential splitting.

65M06, 65M12, 65Z05

bokilv@math.oregonstate.edu (Vrushali A. Bokil)

sakkapla@math.msu.edu (Puttha Sakkaplangkul)

• BibTex
• RIS
• TXT
@Article{IJNAM-15-747, author = {Bokil , Vrushali A. and Sakkaplangkul , Puttha}, title = {Construction and Analysis of Weighted Sequential Splitting FDTD Methods for the 3D Maxwell's Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {15}, number = {6}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12608.html} }
TY - JOUR T1 - Construction and Analysis of Weighted Sequential Splitting FDTD Methods for the 3D Maxwell's Equations AU - Bokil , Vrushali A. AU - Sakkaplangkul , Puttha JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 747 EP - 784 PY - 2018 DA - 2018/08 SN - 15 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12608.html KW - Maxwell’s equations, Yee scheme, Crank-Nicolson method, operator splitting, weighted sequential splitting. AB -

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.

Vrushali A. Bokil & Puttha Sakkaplangkul. (2019). Construction and Analysis of Weighted Sequential Splitting FDTD Methods for the 3D Maxwell's Equations. International Journal of Numerical Analysis and Modeling. 15 (6). 747-784. doi:
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