The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell’s equations

doi:10.1016/j.jcp.2013.02.040

Journal of Computational Physics

Volume 243, 15 June 2013, Pages 344-364

https://doi.org/10.1016/j.jcp.2013.02.040 Get rights and content

Abstract

In this paper we develop a new spatial fourth-order energy-conserved splitting finite-difference time-domain method for Maxwell’s equations. Based on the staggered grids, the splitting technique is applied to lead to a three-stage energy-conserved splitting scheme. At each stage, using the spatial fourth-order difference operators on the strict interior nodes by a linear combination of two central differences, one with a spatial step and the other with three spatial steps, we first propose the spatial high-order near boundary differences on the near boundary nodes which ensure the scheme to preserve energy conservations and to have fourth-order accuracy in space step. The proposed scheme has the important properties: energy-conserved, unconditionally stable, non-dissipative, high-order accurate, and computationally efficient. We first prove that the scheme satisfies energy conversations and is in unconditional stability. We then prove the optimal error estimates of fourth-order in spatial step and second-order in time step for the electric and magnetic fields and obtain the convergence and error estimate of divergence-free as well. Numerical dispersion analysis and numerical experiments are presented to confirm our theoretical results.

Introduction

Maxwell’s equations, which are a set of partial differential equations describing the relation of electric and magnetic fields, have been widely used in computational electromagnetics. They have been playing an important role in many applications such as the radio-frequency, microwave, antennas, aircraft radar, integrated optical circuits, wireless engineering, and the design of CPU in microelectronic, etc. [1], [6], [11], [16], [20], [23].

A very popular numerical method in computing Maxwell’s equations is the finite-difference time-domain (FDTD) scheme, which was first introduced by Yee in [36] and was further developed in [29], [30], etc. Its convergence analysis was carried out in [21]. The scheme has been widely applied to simulate transient electromagnetic wave propagations in a broad range of practical problems with perfectly electric conducting (PEC) boundary conditions or absorbing boundary conditions (ABC). However, the FDTD scheme is only conditionally stable and leads to impractical computational cost and memory requirement in many real-world applications. ADI or fractional step methods have been successful to solve parabolic-type PDEs [5], [8], [9], [10], [18], [25], [26], [27], [35]. The ADI technique has also been applied to overcome the computational complexities and costs in the computation of Maxwell’s equations (see [2], [13], [15], [24], [28], [32], [33], [40]). Holland [15] studied an ADI method combined with Yee’s scheme for the two-dimensional problems. However, the proposed scheme was difficult to obtain the unconditional stability property for Maxwell’s equations in three dimensions. Namiki [24] and Zheng et al. [40] proposed unconditionally stable ADI-FDTD schemes for the two and three dimensional Maxwell’s equations with an isotropic and lossless medium, which were further analyzed in [14], [38], [39]. Diamanti and Giannopoulos [7] discussed the performance of ADI-FDTD sub-grids when implemented into the traditional FDTD method. Wang et al. [32] and Welfert [33] analyzed the iterated ADI-FDTD schemes for solving Maxwell curl equations. Combining the splitting technique with the staggered grids, two splitting FDTD schemes (S-FDTD I&II) were proposed [13] for Maxwell’s equations in two dimensions, which are unconditionally stable and have been applied to solve a scattering problem with PML boundary condition successfully.

On the other hand, the energies of the electromagnetic waves keep constant for all time during the propagation of electromagnetic waves in a lossless medium without sources. These physical identities explain conservations of the electric and magnetic energy during the propagation of electromagnetic waves. It thus is significantly important to physically keep the invariance of energy in time, for developing efficient numerical schemes in computation of Maxwell’s equations and specially in a long term computation of electromagnetic fields. The previous ADI and splitting schemes are unconditionally stable and effective for high dimensional problems, but they usually break the property of energy conservations. Based on the Yee’s grid and splitting technique, Chen et al. [4] first proposed two energy-conserved splitting FDTD schemes (EC-S-FDTD I&II). It was proved both theoretically and numerically that the EC-S-FDTD I&II schemes are energy-conserved and unconditionally stable and the EC-FDTDII scheme is of second order convergence in both time and space steps.

Although the second-order schemes have been widely used with a great deal of success, they are usually efficient only for geometries of moderate electrical size. For computing large scale problems, for problems requiring long-time integration, or for problems of wave propagations over longer distances, it has led to the development of higher-order schemes which produce smaller dispersion or phase errors for a given mesh resolution. Based on the staggered grids, the fourth-order explicit schemes were developed for solving Maxwell’s equations in [12], [17], [31], [34], [37], [41], etc. The use of one-sided high-order difference or extrapolation/interpolation numerical boundary schemes were provided to be suitably accurate relative to the interior differences. However, the explicit higher-order schemes are conditionally stable and lead to prohibitive requirements of computational memory and computational cost. Thus, developing spatial fourth-order energy-conserved S-FDTD schemes for Maxwell’s equations is very important and challenging, which will provide to satisfy discrete energy conservations, unconditional stability, non-dissipativity, and higher-order accuracy.

In this paper, we propose the spatial fourth-order energy-conserved splitting FDTD scheme (i.e. EC-S-FDTD-(2,4)) with fourth-order accuracy in space and second-order accuracy in time. We apply a second-order time-step splitting technique, leading to a three-stage time-splitting algorithm for Maxwell’s equations. At each stage, on the Yee’s staggered grid, we approximate the spatial differential operators by the spatial fourth-order difference operators obtained by a linear combination of two central differences, one with a spatial step and the other with three spatial steps. This obtains the spatial fourth-order scheme in the strict interior nodes of the domain. One important issue is to construct the numerical boundary difference schemes to be energy conservative and high-order relative to the interior difference schemes. It is because the high-order difference operators often have a large spatial stencil which cannot be used in the near boundary nodes. The one-sided differences and extrapolation/interpolation numerical boundary schemes normally break the property of energy conservations near the boundary. Appropriate energy-conserved numerical boundary difference schemes can be difficult to obtain, and this leads to a challenge of constructing energy-conserved higher-order S-FDTD schemes. We propose to construct the spatial fourth-order near boundary differences over the near boundary nodes by using the PEC boundary conditions, original equations and Taylor’s expansion, which ensure the each-stage schemes to preserve the conservations of energy and to have fourth-order accuracy. The proposed EC-S-FDTD-(2,4) scheme has the significant properties that are energy-conserved, unconditionally stable, non-dissipative, high-order accurate, and computationally efficient. We strictly prove that the scheme satisfies energy conversations and is unconditionally stable. We analyze theoretically the convergence of the scheme by using the energy method and obtain the optimal-order error estimates of $O (Δ t^{2} + Δ x^{4} + Δ y^{4})$ in the discrete $L_{2}$ -norm for the approximations of the electric and magnetic fields. Further, the divergence-free convergence is analyzed and we obtain the error estimate of the approximation of divergence-free. Numerical dispersion analysis verifies that the proposed scheme is non-dissipative. Numerical experiments show that the proposed scheme preserves energy conservations and has fourth-order accuracy in space and second-order in time. We also test numerical divergence-free and the divergence-free is of second order convergence in time step and of fourth-order convergence in spatial step.

The paper is organized as follows. In Section 2, the Maxwell’s equations are introduced and the spatial fourth-order EC-S-FDTD scheme is proposed. In Section 3, we prove the property of energy conservations. The truncation error and convergence are analyzed in Section 4. Numerical dispersion analysis and numerical experiments are presented in Section 5. Finally, some conclusions are addressed in Section 6.

Section snippets

Maxwell’s equations and the spatial fourth-order EC-S-FDTD scheme

We first introduce Maxwell’s equations and give the two-dimensional transverse electric (TE) equations. We then present our spatial fourth-order energy-conserved splitting finite difference time domain scheme in this section.

Energy conservation in the discrete form

In this section, we will prove the spatial fourth-order EC-S-FDTD scheme to satisfy two energy conservations in the discrete form. Let the discrete $L^{2}$ -norms on the staggered grids be $‖ U ‖_{E_{x}}^{2} = \sum_{i = 0}^{I - 1} \sum_{j = 0}^{J - 1} {|U_{i + \frac{1}{2}, j}|}^{2} Δ x Δ y, ‖ V ‖_{E_{y}}^{2} = \sum_{i = 0}^{I - 1} \sum_{j = 0}^{J - 1} {|V_{i, j + \frac{1}{2}}|}^{2} Δ x Δ y,$ $‖ W ‖_{H}^{2} = \sum_{i = 0}^{I - 1} \sum_{j = 0}^{J - 1} {|W_{i + \frac{1}{2}, j + \frac{1}{2}}|}^{2} Δ x Δ y, ‖ F ‖_{E}^{2} = ‖ U ‖_{E_{x}}^{2} + ‖ V ‖_{E_{y}}^{2},$ for grid functions $F = (U, V)$ over electric field mesh and W over magnetic field mesh.

For analyzing energy conservations, we first list Lemma 3.1 [22].

Lemma 3.1

For $p ⩾ 1$ , let ${a_{k}}_{k = 1}^{p}$ and ${b_{k}}_{k = 0}^{p}$ be two

Convergence analysis

In this section, we will analyze error estimates of the proposed scheme (2.42), (2.43), (2.44), (2.45), (2.46), (2.47), (2.48), (2.49), (2.50), (2.51), (2.52), (2.53), (2.54), (2.55). In order to do it, we first introduce the (2,4)-order implicit Crank–Nicolson scheme and an equivalent splitting scheme and analyze their truncation errors.

The (2, 4)-order implicit Crank–Nicolson scheme for the Maxwell’s equations can be written as that for the strict interior nodes $\frac{E_{x_{i + \frac{1}{2}, j}}^{n + 1} - E_{x_{i + \frac{1}{2}, j}}^{n}}{Δ t} = \frac{1}{2 ∊} Λ_{y} \{H_{z_{i}}\}$

Numerical experiments

We first take the numerical dispersion analysis for our EC-S-FDTD-(2,4) scheme. Consider the two-dimensional Maxwell’s equations’ solution $E_{α, β}^{n} = E_{0} ξ^{n} e^{- i (k_{x} α Δ x + k_{y} β Δ y)}, H_{α, β}^{n} = H_{z 0} ξ^{n} e^{- i (k_{x} α Δ x + k_{y} β Δ y)},$ where $ξ$ is the increasing factor and the $k_{x}$ and $k_{y}$ are wave numbers along the x-axis and y-axis. By computing, we have the equation of factor $ξ$ $(ξ - 1) (d_{0} ξ^{2} + 2 d_{1} ξ + d_{0}) = 0$ where $d_{0}$ and $d_{1}$ are: $d_{0} = {(1 - \frac{Δ t^{2}}{4 μ ∊} v_{y}^{2})}^{2} + \frac{Δ t^{2}}{μ ∊} (v_{y}^{2} + u_{x}^{2}) + \frac{Δ t^{4}}{16 μ^{2} ∊^{2}} v_{y}^{4} + \frac{Δ t^{4}}{2 μ^{2} ∊^{2}} u_{x}^{2} v_{y}^{2} + \frac{Δ t^{6}}{μ^{3} ∊^{3}} u_{x}^{2} v_{y}^{4},$ $d_{1} = - {(1 - \frac{Δ t^{2}}{4 μ ∊} v_{y}^{2})}^{2} + \frac{Δ t^{2}}{μ ∊} (v_{y}^{2} + u_{x}^{2}) + \frac{Δ t^{4}}{16 μ^{2} ∊^{2}} v_{y}^{4} + \frac{Δ t^{4}}{2 μ}$

Conclusion

In this paper, we proposed the spatial fourth-order energy-conserved splitting finite-difference time-domain scheme (i.e. EC-S-FDTD-(2,4)) for Maxwell’s equations. One important issue is to construct the numerical boundary difference schemes to be energy conservative and high-order relative to the interior difference schemes. The one-sided differences and extrapolation/interpolation numerical boundary schemes normally fail to satisfy energy conservations. At each stage, we proposed the

Acknowledgments

This work was supported by Natural Sciences and Engineering Research Council of Canada. The authors thank the referees and the Associate editor for their suggestions which have helped to improve the paper.

References (41)

M.A. Botchev et al.
Application of the operator splitting to the Maxwell equations with the source term
Appl. Numer. Math.
(2009)
N. Diamanti et al.
Implementation of ADI-FDTD subgrids in ground penetrating radar FDTD models
J. Appl. Geophys.
(2009)
C. Du et al.
An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media
J. Comput. Phys.
(2010)
L. Gao et al.
The splitting-difference time-domain methods for Maxwell’s equations in two dimensions
J. Comput. Appl. Math.
(2007)
A. Nachman
A brief perspective on computational electromagnetics
J. Comput. Phys.
(1996)
E. Turkel et al.
On the construction of a high order difference scheme for complex domains in a Cartesian grid
Appl. Numer. Math.
(2000)
B.D. Welfert
Analysis of iterated ADI-FDTD schemes for Maxwell curl equations
J. Comput. Phys.
(2007)
Z. Xie et al.
An explicit fourth-order staggered finite-difference time-domain method for Maxwell’s equations
J. Comput. Appl. Math.
(2002)
A. Yefet et al.
A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell’s equation
J. Comput. Phys.
(2001)
T.T. Zygiridis et al.
Optimized three-dimensional FDTD discretizations of Maxwell’s equations on Cartesian grids
J. Comput. Phys.
(2007)

H. Ammari et al.

Approximate boundary conditions for patch antennas mounted on thin dielectric layers

Commun. Comput. Phys.

(2006)

L.M. Brekhovskikh

Waves in Layered Media

(1980)

W. Chen et al.

Energy-conserved splitting FDTD methods for Maxwell’s equations

Numer. Math.

(2008)

C.G. Crandall et al.

The method of fractional steps for conservation laws

Numer. Math.

(1980)

D.B. Davidson

Computational Electromagnetics for RF and Microwave Engineering

(2005)

J. Douglas et al.

Improved accuracy for locally one-dimensional methods for parabolic equations

Math. Models Methods Appl. Sci.

(2001)

J. Douglas et al.

On the numerical solutions of heat conduction problems in two and three space variables

Trans. Am. Math. Soc.

(1956)

L. Eyges

The Classical Electromagnetic Fields

(1972)

J. Fang

Time domain finite difference computation for Maxwell’s equations

S. Gedney et al.

Perfectly matched layer media with CFS for an unconditional stable ADI-FDTD method

IEEE Trans. Antennas Propag.

(2001)

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The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell’s equations

Abstract

Introduction

Section snippets

Maxwell’s equations and the spatial fourth-order EC-S-FDTD scheme

Energy conservation in the discrete form

Convergence analysis

Numerical experiments

Conclusion

Acknowledgments

Appl. Numer. Math.

J. Appl. Geophys.

J. Comput. Phys.

J. Comput. Appl. Math.

J. Comput. Phys.

Appl. Numer. Math.

J. Comput. Phys.

J. Comput. Appl. Math.

J. Comput. Phys.

J. Comput. Phys.

Approximate boundary conditions for patch antennas mounted on thin dielectric layers

Commun. Comput. Phys.

Waves in Layered Media

Energy-conserved splitting FDTD methods for Maxwell’s equations

Numer. Math.

The method of fractional steps for conservation laws

Numer. Math.

Computational Electromagnetics for RF and Microwave Engineering

Improved accuracy for locally one-dimensional methods for parabolic equations

Math. Models Methods Appl. Sci.

On the numerical solutions of heat conduction problems in two and three space variables

Trans. Am. Math. Soc.

The Classical Electromagnetic Fields

Time domain finite difference computation for Maxwell’s equations

Perfectly matched layer media with CFS for an unconditional stable ADI-FDTD method

IEEE Trans. Antennas Propag.