电磁波动方程新守恒及对差分格式的分析

NEW CONSERVATION OF ELECTROMAGNETIC WAVE EQUATIONS AND ITS ANALYSIS ON FINITE DIFFERENCE SCHEME

本文研究了具有周期性边界条件的二阶电磁波动方程的守恒性,推出了在H^1、H^2和H^3半范数意义下的恒等式,证明了这类波动方程具有与电磁场旋度的L^2范数有关的新守恒性,并指明了这些恒等式与一般形式的麦克斯韦方程恒等式之间的关系.在此基础上,分析了波动方程的隐式中心差分方法(CN格式),给出了差分格式在离散H^1、H^2和H^3半范数下的数值恒等式和误差分析,证明了CN格式保持新守恒性和超收敛性.数值实验验证了波动方程的新守恒性和对CN格式的数值分析.

Energy conservation of the second order wave equations for electric and magnetic fields with periodic boundary conditions （PBC） is studied and new identities for the fields in terms of H^1 , H^2and H^3 semi norms are derived. It is proved that the electric or magnetic field with PBC is conserved with respect to the H^1 , H^2and H^3 norms. It is found the identities of wave equation are equivalent to those of the general form of Maxwell equations respectively, and that the identities in terms of the H^1 , H^2 and H^3 semi norms reflect that the curls, second and third curls of the electromagnetic fields are conserved in terms of their magnitudes. Based on the new identities, the Crank-Nicolson （CN） scheme for the wave equation is analyzed on stability and convergence. It is proved that the CN scheme is conserved and second order convergent in the discrete H^1 , H^2and H^3 norms. Numerical experiments are oresented to confirm the new conservation of the wave equation and the analysis on the CN scheme.