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Modified Splitting FDTD Methods for Two-Dimensional Maxwell’s Equations

, Shouhui Zhai
Published: 22 February 2017
Mathematical Problems in Engineering , Volume 2017, pp 1-11; doi:10.1155/2017/6063176

Abstract: In this paper, we develop a new method to reduce the error in the splitting finite-difference method of Maxwell’s equations. By this method two modified splitting FDTD methods (MS-FDTDI, MS-FDTDII) for the two-dimensional Maxwell equations are proposed. It is shown that the two methods are second-order accurate in time and space and unconditionally stable by Fourier methods. By energy method, it is proved that MS-FDTDI is second-order convergent. By deriving the numerical dispersion (ND) relations, we prove rigorously that MS-FDTDI has less ND errors than the ADI-FDTD method and the ND errors of ADI-FDTD are less than those of MS-FDTDII. Numerical experiments for computing ND errors and simulating a wave guide problem and a scattering problem are carried out and the efficiency of the MS-FDTDI and MS-FDTDII methods is confirmed.1. IntroductionThe finite-difference time-domain (FDTD) method for Maxwell’s equations, which was first proposed by Yee (see [1], also called Yee’s scheme) in 1966, is a very efficient numerical algorithm in computational electromagnetism (see [2]) and has been applied in a broad range of practical problems by combining absorbing boundary conditions (see [3–7] and the references therein). It is well known from [8] that the Yee Scheme is stable when time and spatial step sizes (, , and for 2D case) satisfy the Courant-Friedrichs-Lewy (CFL) condition , where is the wave velocity. To overcome the restriction of the CFL condition there are many research works on this topic; for example, see [9–17] and the references therein. In [15], two unconditionally stable FDTD methods (named as S-FDTDI and S-FDTDII) were proposed by using splitting of the Maxwell equations and reducing of the perturbation error, where S-FDTDII, based on S-FDTDI (first-order accurate), is second-order accurate and has less numerical dispersion (ND) error than S-FDTDI. However, the second convergence of S-FDTDII was not proved by the energy method.In this letter, by introducing a new method to reduce the error caused by splitting of equations [15] (other methods of reducing perturbation error caused by splitting of differential equations can be seen in [18]), we propose two modified splitting FDTD methods (called MS-FDTDI and MS-FDTDII) for the 2D Maxwell equations. It is proved by the energy method that MS-FDTDI with the perfectly electric conducting boundary conditions is second-order convergent in both time and space. By Fourier method we derive the amplification factors and ND relations of MS-FDTDI and MS-FDTDII. Then, we prove that these two methods are unconditionally stable and that MS-FDTDI has less ND errors than S-FDTDII (or ADI-FDTD [10, 11]). Numerical experiments to compute numerical dispersion errors and convergence orders and to simulate a scattering problem are carried out. Computational results confirm the analysis of MS-FDTDI and MS-FDTDII.2. Modified Splitting FDTD Method for the Maxwell Equations2.1. Maxwell EquationsConsider the two-dimensional Maxwell equations in a lossless and homogeneous medium:where and are the electric permittivity and magnetic permeability of the medium and and for and denote the electric and magnetic fields, respectively. We assume that the spatial domain is surrounded by perfectly electric conductor (PEC). Then the PEC boundary condition below is satisfied:where denotes the boundary of and is the outward normal vector on . The initial conditions are assumed to be where .2.2. Partition of the Domains and NotationsLet be partitioned as Yee’s staggered grids [1]: , and let be divided into equidistant subintervals, , wherewhere and are the spatial step sizes, is the time increment, and , , and are positive integers. For a function and or , we define2.3. Modified Splitting FDTD MethodsDenote by and the approximations to , and , respectively, where, and in what follows, , . Based on the S-FDTDII scheme (see [15]) and the idea of reducing the splitting error, we propose a modified splitting FDTD method (called MS-FDTDI) for (1)–(3).Stage 1. Stage 2. The boundary conditions for (6)–(8) obtained from (2) arewhere or , , .The initial values for (6)–(8) are , and In the implementation of MS-FDTDI, Stage 1 (or Stage 2) can be reduced into a tridiagonal system of linear equations for with (or with ) and a formula for (or ), which can be solved directly.Remark 1. (1) In order to see the difference between MS-FDTDTI and the S-FDTDII method in [15], we give the equivalent forms of the two methods:where (11)–(13) with being the equivalent form of S-FDTDII (Stage 1 of S-FDTDII is the same as (6); Stage 2 of S-FDTDII is (7)-(8) with the last term on the right hand side of (7) removed); (11)–(13) with the case is the equivalent form of MS-FDTDI.By these forms we see that MS-FDTDI is different from the S-FDTDII and ADI-FDTD methods (see [10, 11], where splitting of the equations is not used; however, the equivalent form of S-FDTDII is the same as that of 2D ADI-FDTD).(2) MS-FDTDI has similar perturbation term as the D’yakonov scheme (see [19]). The equivalent form of this scheme is (11)–(13) with and the perturbation term on the right hand side of (12) being removed. In the comparison of these equivalent forms we see that the perturbation term and its location of MS-FDTDI are different from those of the D’yakonov’s scheme. This implies that they are different.Remark 2. Based on S-FDTDII, we propose another modified splitting FDTD method (denoted by MS-FDTDII).Stage 1 of MS-FDTDII.Stage 2 of MS-FDTDII.The boundary and initial conditions of MS-FDTDII are the same as MS-FDTDI.The equivalent form of MS-FDTDII is (11)–(13) withBy these equivalent forms we see that MS-FDTDI and MS-FDTDII are of second-order accuracy.3. Analysis of Stability and Numerical Dispersion ErrorIn this section we first derive the amplification factors and numerical dispersion (ND) relations of MS-FDTDI and MS-FDTDII and then we analyze the stability and ND error.3.1. Stability AnalysisLet the trial time-harmonic solution of the Maxwell equations be where is the unit of complex numbers, , , and are the amplitudes, and are the wave numbers along the -axis and -axis, and is the amplification factor.Substituting the above expressions into the equivalent form of MS-FDTDI and evaluating the determinant of the coefficient matrix of the resulting system of equations for , , and , we get a quadratic equation of . Solving this equation yields the amplification factors for MS-FDTDI:where the coefficients areThe modulus of or is equal to one, implying that MS-FDTDI is unconditionally stable and nondissipative.Similarly, we obtain the amplification factors of MS-FDTDII:where is the same as that in (20), and is That implies that MS-FDTDII is also unconditionally stable and nondissipative.Remark 3. The amplification factors of S-FDTDII, which are the same as those of ADI-FDTD (the derivation is seen in [15]), arewhere is the same as that in (20), and is 3.2. Numerical Dispersion AnalysisLet be the wave speed. Substituting into (20), we obtain the ND relation of MS-FDTDI:where and are defined under (20).Similarly, the ND relation of MS-FDTDII isRemark 4. The ND relation of S-FDTDII is the same as that of ADI-FDTD (see [15]), which isBy using the Taylor expansions of and and the continuous dispersion relation: , we derive the main truncation errors of the ND relations of MS-FDTDI, MS-FDTDII, and S-FDTDII, denoted by , , and , which areBy the second and third terms of truncation errors we see that , implying that the ND error of S-FDTDII or ADI-FDTD is less than that of MS-FDTDII. Noting that we obtain that the ND error of MS-FDTDI is less than that of S-FDTDII (or ADI-FDTD).4. Error Estimates and Convergence of MS-FDTDILet and , where with and denote the values of the exact solution of the Maxwell equations (1)–(3) and with and denote the solution of the MS-FDTDI scheme (6)–(8).Subtracting the equivalent form of MS-FDTDI (11)–(13) from the discretized Maxwell equations (whose form is like (11)–(13) with extra truncation errors), we obtain the following error equations:where with and are the truncation errors, which can be derived by using Taylor formula and discretizing of Maxwell equations. These local truncation error terms are bounded by where and is a constant dependent on norms of the derivatives of the solution of (1)–(3).Multiplying both sides of (31) by , , and , respectively, and applying the summation by parts and the Schwarz inequality we havewhere for and Moreover, if the initial conditions and step sizes satisfythen, by the discrete Growall’s lemma, we haveRemark 5. (1) The convergence of MS-FDTDI requires that with , which is weaker than Courant stability condition: .(2) By the similar method to the above it can not be proved that MS-FDTDII is convergent since the perturbation terms in this scheme are not controlled.5. Numerical ExperimentsWe do some experiments to compute the ND errors of MS-FDTDI and MS-FDTDII, to solve a wave guide problem, and to simulate a scattering problem by the two methods.5.1. Computation of Numerical Dispersion ErrorsLet be the wave length, , and be the number of points per wavelength, and is a multiple of the CFL number (CFL number equals in this case); is the wave propagation angle. Then, by , , , , and the expressions of and (defined in Section 3.1), we see that the amplification or stability factor is a function of , , and ; that is, .The ND errors of MS-FDTDI, MS-FDTDII, and S-FTTDII are computed by the following formula (see [20]): where and denote the imaginary and real parts of the amplification factor . We plot the normalized phase velocity with respect to , and (see Figures 1-2).Figure 1: Normalized phase velocities of MS-FDTDI, MS-FDTDII, and S-FDTDII against wave propagation angle with and .Figure 2: Normalized phase velocities of MS-FDTDI, MS-FDTDII, and S-FDTDII against numbers of points per wavelength with and Figure 1 shows the variation of against the wave propagation with and for MS-
Keywords: Wave Propagation / numerical dispersion / Fourier method / wave guide / Ms Fdtdii / Method of Maxwell

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