摘要
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.
作者
曹敏敏
高理平
郭会
Cao Minmin;Gao Liping;Guo Hui.(School of Science, China University of Petroleum,266580, Qingdao, Shandong, Chin)
出处
《山东师范大学学报(自然科学版)》
CAS
2018年第2期139-149,共11页
Journal of Shandong Normal University(Natural Science)
基金
山东省自然科学基金资助项目(ZR2014AM029)
中央高校基本科研业务费(16CX02017A
18CX05003A)
关键词
麦克斯韦方程
波动方程
有限差分
旋度
守恒
收敛
Maxwell equations
wave equation
finite difference method
curl
conservation
convergence