Optimal error estimates and energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell's equations

TL;DR

This paper establishes optimal error estimates and energy conservation properties for the ADI-FDTD scheme applied to 3D Maxwell's equations, demonstrating its stability and divergence-preserving features.

Contribution

It introduces two new discrete energy norms and energy identities, proving the unconditional stability and divergence preservation of the ADI-FDTD scheme for Maxwell's equations.

Findings

Optimal second-order error estimates in space and time.

Approximate divergence preservation property.

Unconditional stability under four discrete energy norms.

Abstract

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain (ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell equations. Precisely, for the case with a perfectly electric conducting (PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete $H^1$-norm for the ADI-FDTD scheme and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero then the discrete $L^2$-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time. A key ingredient is two new discrete energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell equations introduced in this paper. Furthermore, we prove that, in addition to two known discrete energy identities which are second-order in time perturbations of two known energy conservation laws, the ADI-FDTD scheme also satisfies two new discrete energy identities which are second-order in time perturbations of the two new energy conservation laws. This means that the ADI-FDTD scheme is unconditionally stable under the four discrete energy norms. Experimental results are presented which confirm the theoretical results.