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

TL;DR

This paper introduces a unified framework for implicit FDTD methods using fundamental schemes, enhancing efficiency and stability through generalized matrix formulations and splitting techniques.

Contribution

It develops a generalized formulation of fundamental schemes for implicit FDTD, enabling more efficient and stable algorithms with broad applicability.

Findings

Fundamental schemes unify various implicit FDTD methods.

New implementations show reduced computational costs.

Enhanced stability and efficiency demonstrated through comparisons.

Abstract

This paper presents the generalized formulations of fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain (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 described 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 comparisons of their computation costs and efficiency gains.