The Stability and Numerical Dispersion Study of the ADI-SFDTD Algorithm

Size: px

Start display at page:

Download "The Stability and Numerical Dispersion Study of the ADI-SFDTD Algorithm"

Jeffrey Reynolds
5 years ago
Views:

1 Modern Alied Science Vol. 5, No. 6; Deceber 0 The Stability and Nuerical Dierion Study of the AD-SFDTD Algorith Zongxin Liu(Correonding author ngineering ntitute of ngineering cor, PLA Univerity of Science and Technology Nanjing 007, China & The XXX PLAAF ear of Teting and Training Jiuquan Ganu rovince 73508,China -ail: liuzongxin984077@63.co Yiwang Chen Minitry of Science eearch, PLA Univerity of Science and Technology Nanjing 007, China Tel: ail: chenyiwang00@63.co Xin Xu ngineering ntitute of ngineering cor, PLA Univerity of Science and Technology Nanjing 007, China -ail: xxinn_85@ina.co Xuegang Sun The XXX PLAAF ear of Teting and Training Jiuquan Ganu rovince 73508,China -ail: unxuegang@63.co eceived: Augut 4, 0 Acceted: October 3, 0 Publihed: Deceber, 0 doi:0.5539/a.v5n657 UL: htt://dx.doi.org/0.5539/a.v5n657 Abtract n thi letter, the alternating-direction-ilicit (AD technique i alied to Sylectic finite-difference tie-doain (SFDTD ethod, the curl oerator i endued with two different tyle when doing coutation fro the ( th rogreion to th rogreion. t hold the advantage of both AD-FDTD and SFDTD, not only eliinating the retriction of the Courant-Friedrich-Levy (CFL, but alo holding the inner characteritic of Maxwell equation. The analytical accuracy and efficiency of the rooed ethod i verified good. Keyword: Alternating-direction-ilicit (AD, Sylectic finite-difference tie-doain (SFDTD, Growth atrix, Nuerical dierion. ntroduction FDTD ethod i a very ueful nuerical iulation technique for olving electroagnetic quetion. A we now, the traditional FDTD ethod i baed on the exlicit finite-difference algorith, hence, it i liited by Courant-Friedrich-Levy (CFL tability condition. n order to eliinate the Courant Friedrich Levy (CFL condition retraint, Unconditionally Table algorith ADL-FDTD( the alternating-direction-ilicit technique finite difference tie doain ha been rooed. But in the coon AD-FDTD ethod, the choice of large Publihed by Canadian Center of Science and ducation 57

2 Modern Alied Science Vol. 5, No. 6; Deceber 0 tie interval lead to ubtantial dierion error that degrade it erforance (A. P. Zhao, 000; F. Zheng et al., 000; Huang Z X et al., 007. ffective tudie (M Kuaf et al., 005; uth F D, 983 revealed that Maxwell equation can be viewed a an infinite dienion Hailton yte. FDTD and AD-FDTD detroy Maxwell equation Sylectic tructure, o they are not good algorith for Maxwell equation nuerical iulation. A good algorith ut hold Maxwell equation Sylectic tructure. n thi aer a novel algorith that bae on SFDTD (ylectic finite difference tie doain and AD ha been rooed. We tranfor Maxwell equation to Hailton equation, and ue ylectic roagation technique diere Hailton equation in tie doain, and ue the AD technique to dicredited Hailton equation curl oerator in atial doain, then, we dicu the AD-SFDTD algorith tability and nuerical dierion yteically, finally we validate the rooed AD-SFDTD forulation by a nuerical exale.. AD-SFDTD ethod. Hailton tranfor of Maxwell equation n a linear, hoogeneou, and iotroic ediu, Maxwell equation can be written a (J. W. Thoa, 995: B (a t D HJ (b t D, B H (c Where, i ediu erittivity and i ediu ereability. n the Hailton yte, Maxwell equation can be written a B H ( B, D t 03 3 B D 3 03 H ( B, D t D Where, Hailton function H ( BD, i defined a The B i defined a H ( BD, ( B B D D J B (3 H( BD, H H H,, B Bx By B z H Where ( x, yz, i defined a B d H H ( B 0 dv d (5 B n the equation (5, i the inection function.. AD-SFDTD ethod Studie revealed that Hailton equation can be tranfored into (6, written a B 0 3 B B ( t L L A B (6 D 0 D D 3 T ( (4 58 SSN SSN 93-85

3 Modern Alied Science Vol. 5, No. 6; Deceber 0 0 z y Where, LA, LB, z x 0 y x n tie doain, fro tie t 0 to tiet, the reult of (6 can be written a B ( ex( ( A B B L L (0 (7 D D For exonential oerator can t be ued to coute, the exonential oerator i aroxiate to (8 by uing ylectic roagation technique. ex[ ( A B] ex( d Bex( c A O( L L L L (8 Where,, ( are ylectic roagation rogreion and order. According to (J. W. Thoa, 995, chooe the uitable roagation ub coefficient { c } and{ d }, it can reerve Maxwell equation inner character. n thi aer, we ue the otiized 5 rogreion and 4 order roagation ub coefficient. For ( Lu 0,( u A, B, =,3,, the exonential oerator ex( L A and ex( L B can be exlicitly exreed a ex( LA 6 L A (9a ex( LB 6 L B (9b There i curl oerator in Factor L A and L B, o in order to get the nuerical reult of Maxwell equation, we ut dicredited the equation in atial doain again. ntroducing the lane wave roagation equation, written a: f (, xyzt,, f0 ex( ji ( xx jyy zzz wnt (0 n the herical coordinate yte: x 0 in co, y 0 inin, co z 0. The oitive direction of i that of the right-handed rotation fro x to y about z axi, the oitive direction of i fro the oitive z axi toward the negative z axi, j wave nuber. n and 0 i the nuerical f (, i j, f( ix, jy, z;( n t indicate the th rogreion aroxiate olution of function f ' cloed-for olution at dicretization oint ( ix, jy, z in the n th tie te. There, very tie te need rogreion to iulate and the tie increent of the th rogreion to th rogreion i t. Alying the AD rincile into, we define two different curl oerator about, ared a and in following. n (,, the indicate ilicit for and indicate exlicit for. 0 z y 0 (a z x 0 y x Publihed by Canadian Center of Science and ducation 59

4 Modern Alied Science Vol. 5, No. 6; Deceber 0 0 z y 0 (b z x 0 y x At th rogreion of n th tie te, in x-direction, the ilicit for i defined in equation (a and the exlicit for i defined in (b. n( / n( / f f n i, j, i, j, n fi, j, xfi, j, O( x (a,i x n/ n/ f f n i, j, i, j, n,i fi, j, xfi, j, O( x (b x n n f i j n n f i j f0 ex{ j [( i xx jy y zz w( n ]} f0 ex{ j [( i xx jy y zz w( n ]} n fi, j, x f i, x n the ae way, y-direction ilicit for i, j f i, j, and exlicit for i, j,, for i, f i, j, and exlicit for i,,,. Subtituting (0 into (a and (b, we can get,z-direction ilicit (3a n( / n( / n/ n/ f f f f n i, j, i, j, i, j, i, j, n n w w fi, j, ex( j ex(,i j,i fi, j, h,i fi, j, x x (3b w Where, h ex( j, ex( xx xx j ex( j in( xx x j x x (4a n the ae way we can get y y y ex( y j ex( j yy in( y j y y (4b ex( zz zz j ex( j in( zz z j z z (4c Subtituting (3a and (3b into (a and (b, we get ( h (5 quation (7 can be written a 60 SSN SSN 93-85

5 Modern Alied Science Vol. 5, No. 6; Deceber 0 B ( D B ex[ ( LA LB] (0 D B ex{ [( LA LA ( LB LB]} (0 ( h D B ex[ ( LA LB]ex[ ( LA LB ] (0 ( h ( h D B ex[ ( LA LB]ex[ ( LA LB] (0 ( h ( h D ex[ B ( L ( h A LB] ( D So the coutation fro the ( th rogreion to th rogreion i broen u into two ub-te: the firt te and the econd te. For D x coonent, we tae n( / n( / d n( / n( / D D B B, j, (7a x,( i, j, x,( i, j,,(,,,(,, 4 ( h z i j y i j a the fit te and n( / n ( / n ( / n / d D D B B j,, (7b x,( i, j, x,( i, j,,(,,,(,, 4 ( h z i j y i j a the econd te. For Bx we can get n( / n( / c n( / n( / B B D D, (8a, j x,(, i j, x,(, i j,,(,,,(,, 4 ( h y i j z i j a the fit te and n( / n ( / n ( / n / c B B D D, (8b, j x,(, i j, x,(, i j, 4 ( h y,(, i j, z,(, i j, d c d A the econd te, where T b, T 4 ( h ( ( x, y, z.for other coonent, the equation can be 4 ( h obtained in the ae way. n (7a and (7b, both ide contain the unnown field coonent on the right hand ide, o the iterative calculation can t be done directly. Subtituting (3b into (7a and (7b, we can get (9a and (9b. n( / n( / d n( / n( / n( / d n( / n( / D D Tx B B D Tx B B, j, h, j, x,( i, j, x,( i, j, z,( i, j, y,( i, j, x,( i, j, z,( i, j, y,( i, j, (9a n ( / n ( / n ( / n ( / n ( / n ( / n/ d d D D Tx B B D Tx B h B j,, j,, x,( i, j, x,( i, j, z,( i, j, y,( i, j, x,( i, j, z,( i, j, y,( i, j, (9b 3. Analyi of tability and nuerical dierion 3. Stability analyi According to (6, growth atrix G can be reented a the roduct of the firt rocedure growth atrix G and the econd rocedure G, written a: G GG (0a d c G ex[ ( LA LB] ex( LB ex( L A (0b h h h (6 Publihed by Canadian Center of Science and ducation 6

6 Modern Alied Science Vol. 5, No. 6; Deceber 0 d c G ex[ ( L L ] ex( L ex( L (0c A B B A h h h indicate the growth factor of the total rocedure, indicate growth factor of the firt rocedure and indicate growth factor of the econd rocedure. According to the rincile of the atrix growth, we obtain that atifie equation (a and atifie equation (b: Where, tr( G 0 (a tr( G 0 (b ( h h tr( G [ ] (a x y z g ( h ( h tr( G [ ] (b x y z g ( h g cd cd dc dc (c i j i j i j i j i ji j i ji j By olving (a and (b, the growth factor of the firt rocedure i and the econd rocedure i (, tr( G j 4 tr( G tr( G j tr( G 4 (, (3b Finally, and yield, which indicate the growth factor of the total rocedure a follow: (4 quation (4 i alway atified, o that the AD-STDTD algorith i unconditionally Table in any cae. 3. Nuerical dierion Now we aue of (a and (b[9]. By adding (a and (b we then get quation (5 can be written a follow: Maing ue of (0, (a and (b, we get Putting the into (6, we obtain ( tr( tr( 0 (3a G G (5 ( [ tr( tr( ] 0 G G (6 ( [4in ( / ] (7 ( h ( tr( G tr( G 4 g [ ] (8 x y z ( h in ( xx yy in ( in ( zz ( h in ( w / g{ [ ]} [ ] 0 ( h ( x ( y ( z ( Thi ugget that nuerical dierion of the AD-SFDTD ethod can be reduced to any degree if aroriate cell are ued. Figure i the noralized hae velocity of different for different FDTD chee, we ee the rooed AD-FDTD ha good erforance. Figure how the noralized error contrat between AD-FDTD and AD-SFDTD, it clearly how that AD-SFDTD i ore efficiency. 4. Concluion n thi aer, a novel algorith that baed on SFDTD and AD technique ha be rooed, the atial dicretization chee curl oerator i endued with two different tyle when doing coutation fro the (9 6 SSN SSN 93-85

7 Modern Alied Science Vol. 5, No. 6; Deceber 0 ( th rogreion to th rogreion. Then, t tability and nuerical dierion ha been analyzed. The reult how that the rooed ethod ha good efficiency and accuracy. eference Huang, Z. X., Sha, W, Wu, X. L., et al. (007 Otial ylectic integrator for nuerical olution of tie-doain Maxwell equation Microwave and Otical Technology Letter coutational lectrodynaic, 49(3, Hirono, T., Lui, W., Sei, S., & Yohiuni, Y. (00. A three-dienional fourth-order finite-difference tie-doain chee uing ylectic integrator roagator. Tran action on Microwave theory and Technique, 49, htt://dx.doi.org/0.09/ ftihar, A., &, Zhang, C. Z.(David (006. Dierion-rror Otiized AD FDTD. Microwave Syoiu Diget MTT-S nternational Kuaf, M. A., Oztora, Y., & Daoud, D. S. (005. Otiized exonential oerator coefficient for ylectic FDTD Method. Microwave and Wirele Coonent Letter, 5, htt://dx.doi.org/0.09/lmwc Naii, T. (999. A new FDTD algorith baed on alternating direction ilicit ethod, Tran. Microw. Theory Tech, 47(0, htt://dx.doi.org/0.09/ uth, F. D. (983. A canonical integration technique. Tran.Nucl Sci, 30, htt://dx.doi.org/0.09/tns Thoa, J. W. (995. Nuerical Partial Differential quation: Finite Difference Method. Berlin, Gerany: Sringer Verlag. Zhao, A. P. (00 Two ecial note on the ileentation of the unconditionally Table AD-FDTD ethod, Microw.Ot.Technol Lett, 33(4, htt://dx.doi.org/0.00/o.095 Zheng, F., Chen, Z., & Zhang, J. (000. Toward the develoent of a three-dienional unconditionally Table finite-difference tie-doain ethod. Tran. Microwave. Theory Tech, 48(9, htt://dx.doi.org/0.09/ Figure. Noralized hae velocity of different for different FDTD chee Publihed by Canadian Center of Science and ducation 63

8 Modern Alied Science Vol. 5, No. 6; Deceber 0 Figure. The noralized nuerical dierion error contrat between AD-FDTD and AD-SFDTD 64 SSN SSN 93-85

A Constraint Propagation Algorithm for Determining the Stability Margin. The paper addresses the stability margin assessment for linear systems

A Constraint Propagation Algorithm for Determining the Stability Margin. The paper addresses the stability margin assessment for linear systems A Contraint Propagation Algorithm for Determining the Stability Margin of Linear Parameter Circuit and Sytem Lubomir Kolev and Simona Filipova-Petrakieva Abtract The paper addree the tability margin aement