7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Reduction of Anisotropy in Numerical Dispersion
in the Explicit Finite-Difference Time-Domain Method with Laplacian
Harune Sekido1,2, Takayuki Umeda1 and Yoshizumi Miyoshi1
1Graduate School of Engineering, Nagoya University,
2
Institute for Space-Earth Environmental Research, Nagoya University
e-mail (speaker): sekido@isee.nagoya-u.ac.jp

The Finite-Difference Time-Domain (FDTD) method is anumerical method for solving the time development ofelectromagnetic fields.[1,2] The time-development
equations used in the FDTD method is obtained byapproximating Maxwell’s equations with the finite
difference of second-order accuracy in both time andspace. A staggered grid (Yee grid) system is used in thespatial differences so that Gauss’s law for both electricand magnetic fields is always satisfied. However, theFDTD method has a disadvantage that numericaloscillations occur in discontinuous waveforms and evenin continuous waveforms with a large slope.
Higher-order finite differences reduce the numericalerror in the phase velocity. The FDTD(2,4) method uses
the fourth-order spatial difference.[3,4] The numericalphase velocity error of FDTD(2,4) is smaller than that of
FDTD(2,2). However, the Courant condition ofFDTD(2,4) method is more restricted than that of
FDTD(2,2). In general, higher-order finite differences inspace with the second-order finite difference in time
make the Courant condition more restrictive, which
requires smaller ∆𝑡 and larger number of time steps.
Recently, FDTD method which relaxes the Courant
condition has been developed.[5] The time-development
equations of this method is derived by adding
third-degree difference terms with coefficients to these ofFDTD(2,4). However, there exists an anisotropy in thenumerical dispersion, which results in numericaloscillations.
In the present study, a new explicit and non-dissipative
FDTD method in two and three dimensions is proposed
for reduction of anisotropy in numerical dispersion andrelaxation of the Courant condition. By adding thethird-degree spatial difference terms including Laplacianwith coefficients to the time-development equations ofFDTD(2,4), a new schemes are developed.
Optimal coefficients are obtained by a brute-force
search of the dispersion relations, which reduces phasevelocity errors averaged over the entire wavenumber
space but satisfies the numerical stability as well. Thenew scheme is stable with large Courant numbers up to𝐶 = 1. The new scheme also has smaller numericalerrors in the phase velocity than conventional FDTD
methods.
Figure 1 shows the results of numerical simulations in
two dimensions. Panels (a) and (b) show the spatial
profile of the magnetic field 𝐵𝑧 contentment with
Sekido23 and the present scheme with 𝐶 = 1 ,respectively. Here, Sekido23 is the previous FDTD
scheme which is proposed in the previous study.[5] Panel(a) shows wave propagation is depend on direction, andthe numerical oscillations are the smallest at |𝑥| = |𝑦|with Sekido23. Panel (b) shows anisotropy in thewaveforms is improved, and numerical oscillations arereduced with the present scheme.
References[1] K. S. Yee, IEEE Trans. Antennas Propag., 14, 3, 302–307 (1966)[2] A. Taflove, IEEE Trans. Electromagn. Compat., 22, 3,191–202 (1980)[3] J. Fang, Ph.D. dissertation, “Time domain finite
difference computations for Maxwell’s equations,”(1989)[4] P. G. Petropoulos, IEEE Trans. Antennas Propag., 42,6, 859–862 (1994)[5] H. Sekido and T. Umeda, IEEE Trans. Antennas
Propag., 71, 2, 1630–1639 (2023)

Figure 1. Spatial profiles of 𝐵𝑧 in two dimensions with 𝐶 = 1: (a) Sekido23; (b) present scheme.