1\C 1 I)J1mptMIll 'betl~flbejlli

Transcription

1 1\C 1 )J1mptMll 'betl~flbejll l~~1hlbd ada]t6gh -or~«ejf. '~illrlf J~..6 '~~~ll!~ 4iJ~ "Mf:i',nl.Nqr2l' ~':l:mj}.i~:tv t.~l '\h Dr. N.Homsup, Abstract n this paper, two high-order FDTD schemes are developed to solve the Maxwell's equations with a bounded domain. The first scheme is the extension of the standard Yee algorithms which rely on the approximation of a partial derivative by the central-difference approximation. The second scheme is based on a spline function. This scheme uses the mesh stencil as used in the standard Yee cells and it is relatively easy to modify an existing code based on the Yee algorithm. Both schemes can be adapted for an unbounded space problem such as a scatter in an unbounded space.. ntroduction The finite-difference time -domain (FDTD) scheme is now widely used in vari- ous applications and the literature on it is extensive [1] - [3]. The main advantages of FDTD- based techniques for solving the Maxwell's equations are simplicity and the ability to handle complex geometry. Derivation from Maxwell's equations is based on a method of approximating derivatives by finite difference and approximating line integrals, surface integrals, and volume integrals by summations. t uses basic arithmetic operations-addition, subtraction, multiplication, and division. Material properties are specified at each grid point. At the interface points of different materials. FDTD can model material through pararneter averaging. The field at each grid point is calculated explicitly using only adjacent field's values at previous time. The disadvantages of the FDTD include the require 1 Associate Professor, Department of Electrical Engineering, Faculty of Engineering, Kasetsart University.

2 ment of a high-speed computer and large amount of memory when the linear dimension of the object is large compared to the wavelength because of the dispersion introduced by the algorithm. Most of FOTO schemes are the second-order accurate; the fourth order accurate-compact scheme was introduced by Turkel & Yefet r4]. They approximate the spatial derivative to the fourth order using an implicit method. Their method requires the inversion of a tridiagonal matrix to find all spatial derivatives and all grid points must be uniform. This paper presents two new FOTO schemes. The first scheme is the extension of the standard Yee scheme to higher-order accuracy. The second scheme is a compact FOTO scheme, whith can be used with a nonunifonn grid. All spatial derivati ves are approximated using a cubic spline method. The difference between the Yee scheme and this scheme is the replacement of the second orders accurate spatial derivatives by a higher order accurate derivatives using a cubic spline interpolation. Thus, the scheme is a fourth order in space but second order in time. This is a reasonable option since the temporal accuracy can be improved by choosing a smaller time step. This increases the work only linearly and does not increase the storage. This higher order scheme in space enables one to choose a coarser mesh, This decreases the work in each space dimension and also decreases the storage.. Finite Difference Scheme To simplify the notation, only the two-dimensional case with a ~ 0 is considered. The only sources for the problem are incident waves. The extension of the method to three space dimensions sources and variable coefficients is straightforward. The TM system is consider and the time is normalized with the speed of light, c. let, ~ ct, and Z ~ V~, where z is a wave impedance. The TM wave equations then becomes: de, d1 dh, d, ijh y (h Z(dH y dx 1 de, -Z dy 1 de, Z dx _ d~) dy (1 ) (2) (3) Then, Yee's difference equations m'e presented in the form that will enable one to generalize them easily. " 1 " -- 8 E (4) O[H.x!i,i_1 l 2 Z Y L i,j-1/2 1-8E " () 8[ H y l :~1/2,J Z x z i-1/2,j 5, Where n+1l2 n+1!2 Z[8 x H v l -8 x H x l, ](6) - J.J Un ( U " '/2,-U.'/2 ")/~x (7) x 1,J l+,j 1,.1 S U~+1/2 1"'t1 n ) ( U. U /~t (8 ) ( The second order accuracy of Yee algorithms can be extended to higher order as shown in Table 1.

3 Order of,.~rder of Denvallve accuracy 1?, 6 f, i\ppro;dmalloll at x~,') ~-c",'ruin:ile< dl noj<:s' - i J L -5U -,11'2 i??,/2 5/2 - -, i ,., "" "" &40 & " " "", Cubic Spline nterpolation Consider an interval, a <; x <; b, and divide this interval as a = X o < Xl <... < X O - 1 < X n ~ b, The objective in cubic splines is to derive the third order polynomial for each interval between knots, as (9) t can be shown that,",( x - X,) j;(xi-1)(x -x) 1-1 ' " (X-X,_) + (Xi) (X - X _ ): Xj _ 1 ~X:S; Xi 1 i 1 f, (X;_) 3 f, (X,) J (X,-X) (X-X ) 6(X i - Xi_) 6(x - X _ ) -l, 1 ( 1 0 ) Eq,(l 0) Contain only two unknown r(xi-1) and f"(x,)- Three-second derivatives can be determined by invoking the condition that the first derivatives at the knots must be continuous: (11 ) Eq, (10) can be differentiated to gve an expansion for the first deri vati ve, f this is done for both the (i-1 th and the i th interval and the two results are set equal according to Eq, (11 ), the following relation results: (x~x ) [f(x'+l )-f(x,)] +.x::~ [f(x'_l )-j(x,)]; 1 1,2....,(n-1) (1 2) 1+1 l1 Eq, (12) is written for all interior knots, (n-1) simultaneous equations result with (n+1) unknown second derivatives, However, this is a natural cubic splines, the second derivatives at the end knot are zero and the problem reduces to (n-1) equations with (n-1) unknowns, n addition, notice that the system of equations will be tridiagonal and is extremely easy to solve, This cubic spline interpolation is used to

4 approximate the partial derivation n the higher-order FDTD scheme. V. Absorbing Boundary Recently, a technique for truncation of the computational domains in finite difference methods was proposed [5]. n this method, which is called by the author's transparent absorbing boundary (TAB), a physical problem to be solved is transformed into a problem for auxiliary fields. These fields are equal to zero at the closed boundary. Since the relationship between the physical fields and their auxiliary counterparts is explicitly known and the former can be found from the latter within the computational domain. V. Computational Results The efficiency of there schemes is tested and compared with the standard Yee scheme, c ~ 1 and Z ~ 1 are chosen for all computation. All cases use the TAB for truncating the computational domain. Consider the simplest mode of propagation in a rectangular cross section waveguide as in Fig.1 x Fig. 1 Unbounded Domain The walls are perfect conductors. So with the boundary and initial conditions: E,(x,y,o) EJo,y,t) 0, Ez(l,y,t) o. The solution is then: Ez(x,y,t) ~ sin(j1l:x)sin(31l:y), sin(j1l:x)sin(3n:y-5rrt). The test problem was computed on a sequence of uniformly refined grid h ~ 1/20, L'.t ~ 1/ ,1 D a Fig. 2 Bounded Domain using TAB n the following tests, the function F(x,y) is chosen as: F(x,y) ~ g(y) ~ 1-J(y-0.5) 2 (13) standard form: Error The errors are determined using the N,N, Where E,.. is the true solution and E, is.j L the finite difference solution. n Fig. 3, the error for each scheme was plotted as a function of time. These results demonstrate that the higher-order scheme has less error than the standard Yee scheme or the spline-based scheme.

5 has an existing code one the standard Yee scheme. ~ - - = References [1] A. Taflove. Computational Electrody namics, Artech House, [2J K.Kunz. R.Luebbers. The Finite Difference Time Domain Method for Fig. 3 The errors for the test problem V. Conclusion This paper presents two higherorder FDTD schemes for numerical solutions to the Maxwell equations. The performance of these was tested for a prohlem in two dimensions. t was show that the methods are robust in so far as numerical stabilities are concerned. These results show that the higher-order scheme has less error than the standard Yee scheme or the spline-based scheme. The higher-order central difference schemes are applying since they can provide higher accuracy on coarse grids and this can sigmficantly impact a CPU time, particularly in 3-dimensions such as the interaction of electromagnetic field 10 microwave oven cavities. The major advantage of these schemes is thai il requires less programming if one already Electromagnetic, CRC Press [3] K.L.Shlager and J.B.Schneider. A Selective Survey of the Finite-Difference Time-Domain Literature, EEE Antenas & Propagation Magazine, Jury [d] E. Turkel and A.Yefet, Fourth Order Accurate Compact mplicit Method for the Maxwell Equations. [5] J.Peny and C.A.Balanis. A Generalized Reflection- Free Domain Truncation Method: Transparent Absorbing Boundary, EEE Transaction on Antennas & Propagation, Vol.dO. No.7, July [6] L.N.Trefethem, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Oxford University Computing Laboratorv p.132.