Fundamental Schemes for Efficient Unconditionally Stable Implicit Finite-Difference Time-Domain Methods

Abstract

eneralized formulations of fundamental schemes for efficient unconditionally stable implicit finite-difference timedomain (FDTD) methods . The fundamental schemes constitute a family of implicit schemes that feature similar fundamental updating structures, which are in simplest forms with most efficient right-hand sides. The formulations of fundamental schemes are presented in terms of generalized matrix operator equations pertaining to some classical splitting formulae, including those of alternating direction implicit, locally one-dimensional and split-step schemes. To provide further insights into the implications and significance of fundamental schemes, the analyses are also extended to many other schemes with distinctive splitting formulae. Detailed algorithms are decribed for new efficient implementations of the unconditionally stable implicit FDTD methods based on the fundamental schemes. A comparative study of various implicit schemes in their original and new implementations is carried out, which includes comparsons of their computation costs and efficiency gains.

Reformulation of the ADI-BPM using a fundamental scheme

“…This leads to quite simple implementation of the algorithm, maintaining the equivalence to the conventional method. Incidentally, coupled equations should be solved for the fundamental FDTDs, so that several field components appear in the right-hand sides of implicit finite-difference equations to be solved [8].…”

“…In this article, we reformulate the ADI-BPM with the help of a fundamental scheme [8] that has originally been developed for the efficient implementation of implicit finite-difference time-domain (FDTD) methods. We first present the formulation, in which derivative-free forms are obtained in the right-hand sides of resultant equations.…”

Gpu-Accelerated Fundamental Adi-FDTD With Complex Frequency Shifted Convolutional Perfectly Matched Layer

“…Note thatũ = 2u and v's serve as temporary auxiliary field variables which do not require additional memory [8]. It can be seen now that the algorithm has its right-hand-sides free of matrix operators A and B .…”

“…Such algorithm is included within a family of fundamental implicit schemes, which feature similar fundamental updating structures that are in simplest forms with most efficient matrix-operator-free right-hand-sides [8]. This leads to fundamental ADI-FDTD, or FADI-FDTD in short, which results in much simpler and more concise update equations than the conventional ADI-FDTD implementation.…”

“…The conventional ADI-FDTD method with CFS-CPML is first cast into the compact matrix form. Based on [8], the matrix form is formulated into FADI-FDTD with its right-hand-sides free of matrix operators, resulting in simpler and more concise update equations. Using Compute Unified Device Architecture (CUDA), we further incorporate the FADI-FDTD with CFS-CPML into the GPU to exploit data parallelism.…”