Discrete singular convolution and its application to computational electromagnetics

TL;DR

This work introduces the Discrete Singular Convolution (DSC) framework as a unified, controllable approach for computational electromagnetics. By constructing delta-type, delta-sequence kernels and employing regularization, the DSC framework enables stable approximations of singular convolutions, discretization, and derivative operations, while accommodating a wide range of boundary conditions. Through a Galerkin-induced collocation formulation, DSC subsumes global, local, Galerkin, collocation, and finite-difference methods, offering a tunable balance between accuracy and computational bandwidth. The authors demonstrate the method on waveguide eigenmode analyses, electrostatic potential problems, and 3D wave propagation, reporting high accuracy and robust performance even for complex geometries and long-time integration. Overall, DSC emerges as a promising, flexible tool for high-precision CEM that can be tailored to problem scales and boundary complexities without significant changes to the underlying code base.

Abstract

A new computational algorithm, the discrete singular convolution (DSC), is introduced for computational electromagnetics. The basic philosophy behind the DSC algorithm for the approximation of functions and their derivatives is studied. Approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels. A systematic procedure is proposed to handle a number of boundary conditions which occur in practical applications. The unified features of the DSC algorithm for solving differential equations are explored from the point of view of the method of weighted residuals. It is demonstrated that different methods of implementation for the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Both the computational bandwidth and the accuracy of the DSC algorithm are shown to be controllable. Three example problems are employed to illustrate the usefulness, test the accuracy and explore the limitation of the DSC algorithm. A Galerkin-induced collocation approach is used for a waveguide analysis in both regular and irregular domains and for electrostatic field estimation via potential functions. Electromagnetic wave propagation in three spatial dimensions is integrated by using a generalized finite difference approach, which becomes a global-finite difference scheme at certain limit of DSC parameters. Numerical experiments indicate that the proposed algorithm is a promising approach for solving problems in electromagnetics.