Residual distribution schemes for Maxwell’s equations

摘要

This paper aims to deliver another approach of solving hyperbolic system of equation into the field of computational electromagnetics, known as the residual distribution (RD) or fluctuation splitting method. The RD scheme fills the interstice between finite-element method (FEM) and finite-volume method (FVM), where its idea of calculating flux residual imitates the FVM, but the numerical solutions within a discretized triangular mesh is interpolated with similitudes to the FEM. It was originally designed as a remedy to capture shock problem in Euler system under compact stencil, but its extension to linear-preserving scheme which is the main cause of this work, capable of giving second-order-accurate results and is congenial to the FEM framework. The RD scheme is always vertex-centered therefore has all the conserved variables E and H located at the same nodal point. This topology evades having both vertex-centered and cell-centered coordination in one single mesh, and permits time-discretization other than the staggered time marching scheme, such as the common backward-time discretization which is unconditionally stable. Another contribution of this work is to suggest that row-mass-lumping of the consistent mass-matrix would not mar too much the second-order-accuracy of LDA-RD scheme, which is an upwind scheme where the mass-matrix an impediment for time-updating during the last decade. A prior reconstruction of the upwind mass-matrix is done before lumping up the mass-matrix so that the time-marching nodal update is consistent with the distribution of mass-matrix. The current work covers 3 basic exercises of electromagnetics, they are 2D radiation problem, 2D scattering problem and 3D waveguide propagation. For the 2D scattering and 3D waveguide problems, the perfect electric conductor (PEC) boundary condition is successfully applied using RD construction.