基于Taylor级数展开定理的高阶FDTD的色散分析

时域有限差分法(FDTD)是计算电磁领域中的一类非常重要的研究工具.而Taylor级数展开定理是构造差分格式的一种重要方法,例如Yee格式采用二阶Taylor格式,Fang格式采用四阶Taylor格式.本文借助于采样定理,详细分析了不同阶Taylor中心差分格式的谱特性以及计算误差,并将任意阶Taylor中心差分格式用于数值求解麦克斯韦方程中,严格导出了稳定性条件和数值色散关系的表达式,引入了新的误差定义来衡量算法的好坏.详细地研究了Courant数、网格分辨率CPW和网格长度比率等因素对于数值色散误差的影响,为基于Taylor差分格式的FDTD算法的研究提供了有用的参考.

Numerical Dispersion of FDTD Based on Taylor's Series Expansion Theorem

Taylor series expansion theorem is a very important method for the construction of finite difference.For ex- ample,2-order Taylor central finite difference is adopted in Yee format and 4-order in Fang format.With the knowledge of linear algebra,general formula for arbitrary order Taylor finite difference(TFD)has been derived.In this paper,the analy- sis of spectral property and computational error of arbitrary order TFD is presented in detail.Then it is applied into the nu- merical solution of Maxwell's equations.Stability condition and numerical dispersion relation are derived.And what's more,a new definition of error is introduced for evaluation of the computation.The effect of Courant number,grid resolution (CPW)and ratio of cell on numerical dispersion is analyzed too.