Optimized operator splitting methods for numerical integration of the time domain Maxwell's equations in computational electromagnetics (CEM) are proposed for the first time. The methods are based on splitting the time domain evolution operator of Maxwell's equations into suboperators, and corresponding time coefficients are obtained by reducing the norm of truncation terms to a minimum. The general high-order staggered finite difference is introduced for discretizing the three-dimensional curl operator in the spatial domain. The detail of the schemes and explicit iterated formulas are also included. Furthermore, new high-order Padé approximations are adopted to improve the efficiency of the proposed methods. Theoretical proof of the stability is also included. Numerical results are presented to demonstrate the effectiveness and efficiency of the schemes. It is found that the optimized schemes with coarse discretized grid and large Courant-Friedrichs-Lewy (CFL) number can obtain satisfactory numerical results, which in turn proves to be a promising method, with advantages of high accuracy, low computational resources and facility of large domain and long-time simulation. In addition, due to the generality, our optimized schemes can be extended to other science and engineering areas directly.
The finite-difference time-domain (FDTD(2, 2)) method [
To improve the numerical dispersion, some high-order space strategies have been put forward. For example, Fang proposed the high-order FDTD(4, 4) method [
In this paper, particularly, we consider optimized operator-splitting methods for numerical solution of the time-dependent Maxwell’s equations. In Section
Maxwell’s equations in an isotropic medium can be rewritten in a matrix form as
Note, solution (
For
Let
When one uses coefficients
For example, using Yee grid [
Here we use Padé The averaged permittivity
Comparison between two kinds of approximation. (a) For
Subcell technique specifies for the Ex field calculation in
The conventional Fourier mode method is used to analyze the stability of the proposed
Applying the
In the three-dimensional (3D) case, the continuous-time discrete-space Maxwell’s equations can be written as
CFLmax and accuracy for different methods.
Method | CFLmax | Accuracy | |
---|---|---|---|
(4, 2) | 1.05 | 0.86 | |
(4, 4) | 0.90 | 0.73 | |
(4, 6) | 0.85 | 0.69 | |
FDTD(2, 2) | 0.71 | 0.58 |
Numerical dispersion curves for (4, 4) scheme and FDTD(2, 2) method.
Remember that we cannot choose the discretized order of
We considered a computational domain of 46 by 46 by 46 cells surrounded by a ten-point PML. A vertical dipole
Comparison of results for FDTD(2, 2) and (4, 4) scheme.
FDTD(2, 2) | (4, 4) | |
---|---|---|
Physical time | 200 ns | 200 ns |
CFL | 0.5 | 0.6 |
Time step | 83.33 ps | 200 ps |
Spatial step | 0.05 m | 0.1 m |
No. of steps | 2400 | 1000 |
Total run time | 184 sec | 131 sec |
Average CPU time/step | 0.0766 sec | 0.1310 sec |
Error | 1.5920 | 0.5107 |
Radiation from a dipole with a ten-point PML. (a) The distribution of the
Next, consider a dielectric sphere illuminated by a plane wave propagating in the
The relative error of computed RCS with different scheme at CFL = 0.5. (a)
As we can see, the (4, 4) scheme is more accurate than the (4, 2) scheme under the same discretized grid and CFL number. When the grid enlarges to
The relative error of computed
The relative error of computed
As indicated in figures, we can come to a conclusion as follows. The smaller apace discretized grid we fix, the higher numerical precision we obtain, no matter what scheme we adopt. With the same spatial discretizated scheme, the higher order of time domain discretized, the higher CFL number we get. The (4, 4) scheme with coarse discretized grid and high CFL number can reach satisfactory numerical results, which in turn proves to be a promising method, with advantages of high accuracy, low computational resources, and facility of large-domain and long-time simulation.
We present optimized operator-splitting methods for numerical solution of the time-dependent Maxwell equations in the time domain. The general high-order staggered finite difference is introduced for approximating the three-dimensional curl operator in the spatial domain. The efficiency of the
This work is supported by the Key National Natural Science Foundation of China (no. 60931002) and Universities of Natural Science Foundation of Anhui Province (no. KJ2011A002).