High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

Abstract

This paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high-order finite difference time-domain (FDTD) schemes of computational electromagnetics (CEM) with material interfaces in one (1D) and two spatial dimensions (2D). By making use of fictitious points, systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage, to arbitrarily high orders of accuracy in principle. While often limited by numerical instability, orders up to 16 in 1D and 2D are achieved. Detailed stability analyses are presented for the present approach to examine the up limit in constructing embedded FDTD methods. As natural generalizations of the high-order FDTD schemes, the proposed derivative matching methods automatically reduce to the standard FDTD schemes when the material interfaces are absent. An interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code. Another feature of the present approach is that it can be robustly implemented with other high accuracy time-domain approaches, such as the multiresolution time-domain (MRTD) method and the local spectral time-domain (LSTD) method, to cope 1 with material interfaces. Numerical experiments on both 1D and 2D problems are carried out to test the convergence, examine the stability, access the efficiency, and explore the limitation of the proposed methods. It is found that operating at their best capacity, the proposed high order schemes are at least a million times more efficient than their fourth order versions in both 1D and 2D. Therefore, it is believed that the proposed hierarchical derivative matching methods are highly accurate, efficient, and robust for CEM.

Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces

“…These problems call for new efficient methods that do not depend on massive local mesh refinement, which does not work for highly oscillatory waves due to the pollution effect [3]. The present work provides a solution to this class of problems on the Cartesian grid by extending the marched interface and boundary (MIB) method [60,61] previously designed for straight or curved interfaces. The concept of secondary fictitious values is introduced to deal with difficult topology where primary fictitious values cannot be solved directly.…”

“…Aforementioned methods have found much success in scientific and engineering applications [6][7][8]15,18,20,[25][26][27][28]30,32,34,39,41,40,42,53,54,[57][58][59]. A possible further direction in the field could be the development of higher order interface methods [20,60,61] which are particularly desirable for problems involving both material interfaces and high frequency oscillations, such as the interaction of turbulence and shock, and high frequency wave propagation in inhomogeneous media [5].…”

“…The MIB was proposed by Zhao and Wei [60] as a systematic higher-order method for electromagnetic wave propagation and scattering in dielectric media. Recently, it has been generalized for solving elliptic equations with curved interfaces by Zhou et al [61].…”

“…Since only lowest order interface jump conditions are repeatedly used in the MIB method, arbitrarily high order convergence can be achieved in principle. For straight interfaces, MIB schemes of up to 16th order have been constructed [60,61]. For lightly curved interfaces, up to 6th order schemes have been demonstrated [61].…”

High-order fractional partial differential equation transform for molecular surface construction

“…It is very difficult to construct second-order convergent methods for this equation in the biomolecular context due to the geometric complexity, complex interface, singular charge sources and geometric singularities [15, 24, 67]. In this work, we make use of the second-order convergent matched interface and boundary (MIB) method [15, 68, 69, 73, 77, 78] to solve Eq. (60).…”