A Real-space Green's Function Method for the Numerical Solution of Maxwell's Equations in Free Space

Victor Minden, Stanford University

A new method for solving the transverse part of the free-space Maxwell’s 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. Due to the singular nature of the propagator, in our method we regularize via convolution with smoothing kernels (B-splines) prior to sampling based on a method due to Beyer and LeVeque. 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 and parallelized using the same observation as made by Vay, Haber and Godfrey, leading to a method that is computationally efficient.

Abstract Author(s): Victor Minden, Boris Lo, Phillip Colella