High-accuracy Yee algorithm based on nonstandard finite differences: new developments and verifications

We previously described a high-accuracy version of the Yee algorithm that uses second-order nonstandard finite differences (NSFDs) and demonstrated its accuracy numerically. We now prove that at fixed frequency and grid spacing h, the leading error term is O(h/sup 6/) versus O(h/sup 2/) for the ordinary Yee algorithm with standard finite differences (SFDs). We numerically verify the superior accuracy of the NSFD algorithm by simulating near-field Mie scattering on a coarse grid and comparing with the SFD one and with analytical solutions. We present an updated stability analysis and show that the maximum time step for the NSFD algorithm is 20% longer than the SFD time step in two dimensions, and 36% longer in three dimensions. Finally, parameters that were previously given numerically are now analytically defined.

[1]  Melinda Piket-May,et al.  A modified FDTD (2, 4) scheme for modeling electrically large structures with high-phase accuracy , 1997 .

[2]  J. B. Cole.  A high-accuracy realization of the Yee algorithm using non-standard finite differences , 1997 .

[3]  K. Yee.  Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media , 1966 .

[4]  J. B. Cole.  A high accuracy FDTD algorithm to solve microwave propagation and scattering problems on a coarse grid , 1995 .

[5]  J. B. Cole.  Application of Nonstandard Finite Differences to Solve the Wave Equation and Maxwell's Equations , 2000 .

[6]  Yen Liu,et al.  Fourier Analysis of Numerical Algorithms for the Maxwell Equations , 1993 .

[7]  Yaowu Liu,et al.  A higher order (2,4) scheme for reducing dispersion in FDTD algorithm , 1999 .

[8]  James B. Cole.  A nearly exact second-order finite difference time-domain wave propagation algorithm on a coarse grid , 1994 .

[9]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[10]  James B. Cole,et al.  High accuracy solution of Maxwell's equations using nonstandard finite differences , 1997 .

[11]  Robert J. Lee,et al.  Reducing the phase error for finite-difference methods without increasing the order , 1998 .

[12]  R. Booton,et al.  Computational Methods for Electromagnetics and Microwaves , 1992 .

[13]  Generalized nonstandard finite differences and physical applications , 1998 .

[14]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[15]  S. C. Hill,et al.  Light Scattering by Particles: Computational Methods , 1990 .

[16]  J. B. Cole,et al.  Finite Difference Time Domain Computation of Light Scattering by Multiple Colloidal Particles , 1998 .

[17]  R. Mickens.  Nonstandard Finite Difference Models of Differential Equations , 1993 .