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

• Published in: 中国科学：数学英文版 2013 (8), p.1705-1726
• Main Author: GAO LiPing ZHANG Bo
• Format: Article
• Language: English
• Subjects: ADI-FDTD; 交错网格; 修改; 最优误差估计; 能量守恒定律; 节约能源; 麦克斯韦方程组
• ISSN: 1674-7283; 1869-1862

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＇s 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 Hi-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 L2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time. The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell＇s equations introduced in this paper. ~rthermore, we prove that, in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws, the ADI-FDTD scheme also satisfies two new discrete modified 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 modified energy norms. Experimental results which confirm the theoretical results are presented.