Energy-preserving local mesh-refined splitting FDTD schemes for two dimensional Maxwell's equations

Abstract

In this paper, we develop and analyze two types of new energy-preserving local mesh-refined splitting finite difference time-domain (EP-LMR-S-FDTD) schemes for two-dimensional Maxwell's equations. For the local mesh refinements, it is challenging to define the suitable local interface schemes which can preserve energy and guarantee high accuracy. The important feature of the work is that based on energy analysis, we propose the efficient local interface schemes on the interfaces of coarse and fine grids that ensure the energy conservation property, keep spatial high accuracy and avoid oscillations and meanwhile, we propose a fast implementation of the EP-LMR-S-FDTD schemes, which overcomes the difficulty in solving unknowns on the “trifuecate structure” of refinement by first solving the values of coarse mesh unknowns and the average values of fine mesh unknowns on a line-structure and then solving the values of fine mesh unknowns and the coarse mesh unknown on an inverted “U-form” structure for each loop. The EP-LMR-S-FDTD schemes can be solved in a series of tridiagonal linear systems of unknowns which can be efficiently implemented at each time step. We prove the EP-LMR-S-FDTD schemes to be energy preserving and unconditionally stable. We further prove the convergence of the schemes and obtain the error estimates. Numerical experiments are given to show the performance of the EP-LMR-S-FDTD schemes which confirm theoretical results.