Towards the Development of A Three-Dimensional SBP-SAT FDTD Method: Theory and Validation

TL;DR

This paper introduces a three-dimensional SBP-SAT FDTD method that improves scalability and stability for electromagnetic simulations, maintaining accuracy with minimal computational overhead.

Contribution

It develops a theoretically stable 3D SBP-SAT FDTD method capable of handling complex problems with multiple mesh blocks and boundary conditions.

Findings

Method is stable and accurate for complex electromagnetic problems.

Minimal runtime and memory overhead compared to traditional FDTD.

Validated through four numerical examples.

Abstract

To enhance the scalability and performance of the traditional finite-difference time-domain (FDTD) methods, a three-dimensional summation-by-parts simultaneous approximation term (SBP-SAT) FDTD method is developed to solve complex electromagnetic problems. It is theoretically stable and can be further used for multiple mesh blocks with different mesh sizes. This paper mainly focuses on the fundamental theoretical aspects upon its three-dimensional implementation, the SAT for various boundary conditions, and the numerical dispersion properties and the comparison with the FDTD method. The proposed SBP-SAT FDTD method inherits all the merits of the FDTD method, which is matrix-free, easy to implement, and has the same level of accuracy with a negligible overhead of runtime (0.13\%) and memory usage (1.2\%). Four numerical examples are carried out to validate the effectiveness of the proposed method.