Time-Domain Numerical Solutions of Maxwell Interface Problems with Discontinuous Electromagnetic Waves (Advances in Applied Mathematics and Mechanics, 2018)
This paper is devoted to time domain numerical solutions of two-dimensional(2D) material interface problems governed by the transverse magnetic(TM) and transverse electric (TE) Maxwell's equations with discontinuous electromagneticsolutions. Due to the discontinuity in wave solutions across the interface, theusual numerical methods will converge slowly or even fail to converge. This calls forthe development of advanced interface treatments for popular Maxwell solvers. Wewill investigate such interface treatments by considering two typical Maxwell solvers– one based on collocation formulation and the other based on Galerkin formulation. Torestore the accuracy reduction of the collocation finite-difference time-domain (FDTD)algorithm near an interface, the physical jump conditions relating discontinuous wavesolutions on both sides of the interface must be rigorously enforced. For this purpose,a novel matched interface and boundary (MIB) scheme is proposed in this work, inwhich new jump conditions are derived so that the discontinuous and staggered featuresof electric and magnetic field components can be accommodated. The resultingMIB time-domain (MIBTD) scheme satisfies the jump conditions locally and suppressesthe staircase approximation errors completely over the Yee lattices. In the discontinuousGalerkin time-domain (DGTD) algorithm – a popular Galerkin Maxwell solver, aproper numerical flux can be designed to accurately capture the jumps in the electromagneticwaves across the interface and automatically preserves the discontinuity inthe explicit time integration. The DGTD solution to Maxwell interface problems is exploredin this work, by considering a nodal based high order discontinuous Galerkinmethod. In benchmark TM and TE tests with analytical solutions, both MIBTD andDGTD schemes achieve the second order of accuracy in solving circular interfaces. Incomparison, the numerical convergence of the MIBTD method is slightly more uniform,while the DGTD method is more flexible and robust.