S-ERKN methods solving Maxwell’s equations in source field under wide range of boundary conditions

For Maxwell’s equations for isotropic homogeneous media in a cubic domain, there have been few methods that can solve almost all problems under various boundary conditions in source field since the boundary conditions and the sources are diverse, by now. Fortunately, in this paper, we succeeded in giving a unified class of methods (called S-ERKN methods) that solve problems under a wide range of boundary conditions in source field. At first, we gave a detailed study on the spatial discretization to have a so-called semi-discrete system. Here we concluded that the boundary conditions had an unsurprising impact on the differentiation matrix (denoted as LU in this paper) but had an unexpected effect on the perturbation term (denoted as F̃U in this paper) of the semi-discrete system. Secondly, we chose the ERKN (Extended Runge–Kutta Nystrom) methods as the temporal solver to improve the temporal order which is normally limited to 2 in many traditional numerical solutions. To make the S-ERKN method (obtained by the spatial and temporal discretization shown above) easy to code at low cost, we studied the fast calculation of the matrix vector product LUV, which is the basic computational unit of the method. In this paper, we also performed the divergence analysis of the methods. The numerical experiments illuminate that the method can be explicit, highly accurate, divergence-free in numerical terms, and of good long-term behavior.