交错网格上ADI-FDTD格式求解三维麦克斯韦方程组的最优误差估计与能量守恒恒等式
Optimal error estimates and energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell's equations
AI总结：
本文针对三维麦克斯韦方程组的交替方向隐式时域有限差分（ADI-FDTD）方法，建立了离散H^1范数下的时空二阶最优误差估计，并证明了近似散度保持性质及两个新的离散能量恒等式。
摘要：
本文关注交替方向隐式时域有限差分（ADI-FDTD）方法的最优误差估计和能量守恒性质，该方法是一种求解三维麦克斯韦方程组的流行格式。具体地，对于完美电导体（PEC）边界条件的情况，我们在离散$H^1$范数下建立了ADI-FDTD格式在空间和时间上的最优二阶误差估计，并证明了近似散度保持性质：如果初始电场和磁场的散度为零，则ADI-FDTD解的离散散度的离散$L^2$范数在空间和时间上以二阶精度近似为零。一个关键要素是两个新的离散能量范数，它们是本文引入的麦克斯韦方程组两个新能量守恒定律的时间二阶扰动。此外，我们证明，除了两个已知的离散能量恒等式（它们是两个已知能量守恒定律的时间二阶扰动）外，ADI-FDTD格式还满足两个新的离散能量恒等式，它们是两个新能量守恒定律的时间二阶扰动。这意味着ADI-FDTD格式在四个离散能量范数下是无条件稳定的。实验结果验证了理论结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
