Vol. 11, No. 2, 2016

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A real-space Green's function method for the numerical solution of Maxwell's equations

Boris Lo, Victor Minden and Phillip Colella

Vol. 11 (2016), No. 2, 143–170

A new method for solving the transverse part of the free-space Maxwell equations in three dimensions is presented. By taking the Helmholtz decomposition of the electric field and current sources and considering only the divergence-free parts, we obtain an explicit real-space representation for the transverse propagator that explicitly respects finite speed of propagation. Because the propagator involves convolution against a singular distribution, we regularize via convolution with smoothing kernels (B-splines) prior to sampling based on a method due to Beyer and LeVeque (1992). We show that the ultimate discrete convolutional propagator can be constructed to attain an arbitrarily high order of accuracy by using higher-order regularizing kernels and finite difference stencils and that it satisfies von Neumann’s stability condition. Furthermore, the propagator is compactly supported and can be applied using Hockney’s method (1970) and parallelized using the same observation as made by Vay, Haber, and Godfrey (2013), leading to a method that is computationally efficient.

Maxwell's equations, Green's function, high order
Mathematical Subject Classification 2010
Primary: 65M12, 65M80
Secondary: 65D05, 65D07, 78M25
Received: 4 May 2015
Revised: 28 January 2016
Accepted: 21 February 2016
Published: 11 August 2016
Boris Lo
Applied Science and Technology
University of California, Berkeley
Berkeley, CA 94720
United States
Victor Minden
Institute for Computational and Mathematical Engineering
Stanford University
Stanford, CA 94305
United States
Phillip Colella
Lawrence Berkeley National Laboratory
Berkeley, CA 94720
United States
Electrical Engineering and Computer Sciences
University of California, Berkeley
Berkeley, CA 94720
United States