This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

Transcription

1 Ths artcle appeared a joral pblshed by Elsever. The attached copy s frshed to the athor for teral o-commercal research ad edcato se, cldg for strcto at the athors sttto ad sharg wth colleages. Other ses, cldg reprodcto ad dstrbto, or sellg or lcesg copes, or postg to persoal, stttoal or thrd party webstes are prohbted. I most cases athors are permtted to post ther verso of the artcle e.g. Word or Tex form) to ther persoal webste or stttoal repostory. Athors reqrg frther formato regardg Elsever s archvg ad mascrpt polces are ecoraged to vst:

2 Appled Nmercal Mathematcs 59 9) Cotets lsts avalable at SceceDrect Appled Nmercal Mathematcs A aalyss of the fte-dfferece method for oe-dmesoal Kle Gordo eqato o boded doma Hode Ha, Zhwe Zhag Departmet of Mathematcal Sceces, Tsgha Uversty, Bejg 184, PR Cha artcle fo abstract Artcle hstory: Receved 4 September 7 Receved revsed form 1 October 8 Accepted 4 October 8 Avalable ole 6 November 8 Keywords: Artfcal Bodary Codto ABC) Uboded doma Eergy method Fast algorthm Dscrete Artfcal Bodary Codto DABC) The mercal solto of the oe-dmesoal Kle Gordo eqato o a boded doma s aalyzed ths paper. Two artfcal bodary codtos are obtaed to redce the orgal problem to a tal bodary vale problem o a boded comptatoal doma, whch s dscretzed by a explct dfferece scheme. The stablty ad covergece of the scheme are aalyzed by the eergy method. A fast algorthm s obtaed to redce the comptatoal cost ad a dscrete artfcal bodary codto DABC) s derved by the Z-trasform approach. Fally, we llstrate the effcecy of the proposed method by several mercal examples. 8 IMACS. Pblshed by Elsever B.V. All rghts reserved. 1. Itrodcto The Kle Gordo eqato arses relatvstc qatm mechacs ad feld theory, whch s of great mportace for the hgh eergy physcsts [15], ad s sed to model may dfferet pheomea, cldg the propagato of dslocatos crystals ad the behavor of elemetary partcles. The oe-dmesoal Kle Gordo eqato s gve by the followg partal dfferetal eqato: h h c m c 4 = f x, t), x R 1, t >, 1.1) where = x, t) represets the wave desty at posto x ad tme t, h s the Plack costat, c ad m are partcle velocty ad partcle mass, respectvely. There are a lot of stdes o the mercal solto of tal ad tal-bodary problems of the lear or olear Kle Gordo eqato. For example, Khalfa ad Elgamal [] developed a mercal scheme based o a fte elemet method for the olear Kle Gordo eqato wth Drchlet bodary codto o a boded doma, whch shows the overflow solto as expected. Dca [6] aalyzed three fte dfferece approxmatos of the tal olear Kle Gordo eqato, showed they are drectly related to symplectc mappgs ad tested the schemes o the travelg wave ad perodc breather problems over log tme tervals. However, whe we wsh to solve the Kle Gordo mercally o a boded doma, these methods wll face essetal dffcltes. Sce the bodedess of the physcal doma or problem, the stadard fte elemet method or fte dfferece method ca t be sed drectly. Ths work was spported by The NSFC Project. No * Correspodg athor. E-mal addresses: hha@math.tsgha.ed.c H. Ha), zhagzhwe@mals.tsgha.ed.c Z. Zhag) /$3. 8 IMACS. Pblshed by Elsever B.V. All rghts reserved. do:1.116/j.apm.8.1.5

3 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) The artfcal bodary codto ABC) method s a powerfl approach to redce the problems o the boded doma to a boded comptatoal doma. I the well-kow paper of Egqst ad Majda [1], absorbg bodary codtos sg Padé approxmato of a psedo-dfferetal operator o a le-type bodary for the wave eqato are derved. Hgdo [,1] developed radato bodary codtos for the mercal modelg of dspersve waves. By specfyg the wavember ad freqecy parameters, the bodary codtos based o compostos of smple frst-order dfferetal operators was got. Hs formlas ca be appled wthot modfcato to hgher-dmesoal problem. Low-order local ABCs may have low accracy, whereas hgh-order local ABCs are sally hard to mplemet becase they typcally volve hgh-order dervatves [13]. The local ABCs may geerate some ophyscal reflecto at the artfcal bodary ad the well-posedess of the resltg trcated tal-bodary problem s stll ope geeral. Nolocal artfcal bodary codtos have the potetal of beg more accrate tha the local oes. Ha ad Zheg [19] obtaed three kds of exact oreflectg bodary codtos for exteror problems of wave eqatos two ad three-dmesoal space by a approach based o Dhamel s prcple. X. Atoe ad C. Besse [] obtaed a oreflectg bodary codtos for the oe-dmesoal Schrödger eqato. X. Atoe, C. Besse ad V. Moysset [3] also geeralzed ther approach to smlate the two-dmesoal Schrödger eqato sg oreflectg bodary codtos. Ha ad Hag [16], Ha, Y ad Hag [18] derved the exact oreflectg bodary codtos for two- ad three-dmesoal Schrödger eqatos. Ha ad Y [17] also derved the exact oreflectg bodary codtos for two- ad three-dmesoal Kle Gordo eqatos. Geerally speakg, the exact bodary codtos reqre more comptatoal cost. I order to overcome ths dsadvatage, a fast algorthm s gve or paper. The orgazato of ths artcle s the followg: I Secto we trodce two artfcal bodares ad fd the artfcal bodary codtos, the redce the orgal problem to a eqvalet problem o the boded comptatoal doma. I Secto 3, a fte-dfferece scheme for the redced problem s gve ad ts stablty ad covergece are aalyzed. A fast algorthm s obtaed Secto 4 to redce the comptatoal cost. I Secto 5, a dscrete artfcal bodary codto DABC) s derved by the Z -trasform approach. Some mercal reslts wll be gve Secto 6 to demostrate the accracy ad effcecy of the proposed methods.. The artfcal bodary codto I ths secto, we stdy the mercal approxmato of a dspersve wave solto x, t), to the eqato wth a sorce term o a boded doma. More precsely, we cosder the followg lear Kle Gordo eqato o R 1 [, T ]: a b = f x, t), x R 1, t [, T ],.1) t= = ϕ x), x R 1,.) t t= = ϕ 1 x), x R 1..3) Here a, b are two real costats. Assme ϕ x), ϕ 1 x) ad f x, t) satsfyg: Sppϕ x) [x l, x r ],Sppϕ 1 x) [x l, x r ] ad Spp f x, t) [x l, x r ] [, T ]. For smplcty of the dedcto, we take x l =adx r = 1. I order to redce the problem.1).3) to a boded comptatoal doma, we trodce two artfcal bodares, Σ r = x, t) x = 1, t T, Σ l = x, t) x =, t T, whch dvde R 1 [, T ] to three parts, D l = x, t) x, t T, D = x, t) x 1, t T, D r = x, t) 1 x, t T. The boded doma D s or comptatoal doma. Cosder the restrcto of x, t) o the boded doma D r. x, t) satsfes, a b =, x, t) D r, Σr = 1, t) g 1 t), t= =, x 1, t t= =, x 1..4).5).6).7)

4 157 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Sce 1, t) s kow, the problem.4).7) s complete, whch caot be solved depedetly. If Σr = 1, t) g 1 t) s gve, the problem above has a qe solto. Let Ux, s) = L x, t) ) = e st x, t) dt, Re s >, deotes the Laplace trasform of the kow solto x, t). By.4).7) t satsfes, s Ux, s) a U xx x, s) b Ux, s) =, 1 < x <,.8) U1, s) = Gs) L g 1 t) ) = e st g 1 t) dt,.9) Ux, s) <, x..1) Eq..8) s a secod-order lear ODE wth costat coeffcet, ts geeral solto s gve by ) ) s b Ux, s) = c 1 s) exp s b x 1) c s) exp x 1). a a The codto.1) mples c s), ad we obta ) s b Ux, s) = c 1 s) exp x 1),.11) a here the roots wth postve real parts are take. The partal dervatve wth respect to x yelds, ) Ux, s) s b = s b c 1 s) exp x 1)..1) a a Combg.11) ad.1) o the artfcal bodary Σ r,wearrveat U1, s) s b = 1 U1, s) = a s U1, s) b U1, s) )..13) a s b By the table of Laplace trasform see page 118 of [14]), we obta ) L 1 = J bt), s b L s U1, s) b U1, s) ) = 1, t) b 1, t). The, by the covolto theorem of Laplace trasforms, from.13) we obta: 1, t) = 1 a = 1 1, t) b a a = 1 a [ J bt bτ ] 1, τ ) ) τ b 1, τ ) dτ 1, t) b a J 1, τ ) t bt bτ ) dτ b a [ J bt bτ ) J bt bτ ) ] 1, τ ) dτ. J bt bτ )1, τ ) dτ We get the artfcal bodary codto of the problem.1).3) o Σ r. Smlarly we ca get the artfcal bodary codto o Σ l,wtht [, T ]. Hece, we ca redce the tal bodary vale problem o the comptatoal doma D. a b = f x, t), x, t) [, 1] [, T ],.14) t= = ϕ x), t t= = ϕ 1 x), x [, 1],.15) 1, t), t) 1, t) a = b, t) a = b [ J bt bτ ) J bt bτ ) ] 1, τ ) dτ,.16) [ J bt bτ ) J bt bτ ) ], τ ) dτ..17)

5 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Let Jx) = J x) J x), where Jx) s a specal fcto. By some basc recrso formlas of Bessel fctos see page 4 of [1]), we have Jx) := J x) J x) = J x) J 1 x)) 1 = J x) J x) ). The artfcal bodary codtos.16),.17) wll have two smple forms: 1, t) 1, t) t a = b Jbt bτ )1, τ ) dτ,.18), t), t) t a = b Jbt bτ ), τ ) dτ..19) Moreover, let Fx) deotes the prmtve of Jx): Fx) = x Js) ds = ) k ) k1 x ) k ) k3 x x. k!k!k ) k!k )!k 3) k= k= Itegratg by parts Eqs..18) ad.19), we obta ather two eqvalet forms of bodary codtos. The ew forms wll brg some coveece for the proof of stablty ad covergece. 1, t), t) 1, t) t a = b, t) t a = b Fbt bτ ) 1, τ ) dτ..) τ Fbt bτ ), τ ) dτ..1) τ Next, we dscss the qeess ad stablty estmate of the redced problem.14).17). Mltply Eq..1) by ad tegrate wth respect to x [, 1] for fxed t, T ], sg the bodary codto.16),.17), we get d 1 1 [ ) ), t), t) a b 1, t) x, t)] dx a B 1, t) ), t) a B, t) ) dt Here 1 = B νt) ) = 1 a, t) f x, t) dx. dνt) dt b a [ J bt bτ ) J bt bτ ) ] ντ ) dτ. We trodce two axlary fctos W 1) x, t) ad W ) x, t), whch satsfy the followg problems, respectvely:.) ad W 1) tt a W 1) xx b W 1) =, x, t) D r, W 1) Σr = 1, t), t T, W 1) t= =, 1 < x <, W 1) t t= =, 1 < x < W ) tt a W ) xx b W ) =, x, t) D l, W ) Σl =, t), t T, W ) t= =, < x <, W ) t t= =, < x <.

6 157 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Throgh a smlar aalyss, we obta, ad 1, t) a B 1, t) ) = d 1 dt, t) a B, t) ) = d 1 dt 1 [ W 1) ) x, t) W 1) ) a x, t) b W 1) x, t) ) ] dx,.3) We trodce Et) ad Ft) as followg, Et) = 1 Ft) = [ ) ), t), t) a b x, t)] dx [ W ) ) x, t) W ) ) a x, t) b W ) x, t) ) ] dx..4) [ W 1) ) x, t) W 1) ) a x, t) b W 1) x, t) ) ] dx [ W ) ) x, t) W ) a x, t) f x, t) ) dx. Combg.).6) ad sg Cachy Schwarz eqalty, we obta, ) b W ) x, t) ) ] dx,.5).6) d Et) Et) Ft), t T. dt Usg the Growall eqalty from eqalty.7) ad otcg that,.7) E) = 1 1 [ ϕ1 x) ) a ϕ 1 x)) b ϕ x) ) ] dx. we get the stablty estmate for the solto of the redced problem.14).17): 1 [ ) ), t), t) a b x, t)] dx 1 [ ϕ1 x) ) a ϕ 1 x)) b ϕ x) ) t ] dx 1 The, we obta the followg stablty estmate of the redced problem.14).17). e t τ f x, τ ) ) dxdτ..8) Theorem.1. The redced problem.14).17) has at most oe solto x, t) o the boded comptatoal doma D,ad x, t) cotosly depeds o the tal vale ϕ x), ϕ 1 x), ad fx, t). From Theorem.1, we kow that the redced problem.14).17) s eqvalet to the orgal problem.1).3). Namely, the solto of the redced problem.14).17) s the restrcto of the solto of orgal problem.1).3) o the boded doma D,vceversa. 3. Aalyss of the dfferece scheme I ths secto, we cosder the fte dfferece approxmato of the redced problem.14).17) o the boded doma D.WedvdethedomaD by a set of les parallel to the x- ad t-axes to form a grd. We wrte h = 1/I ad τ = T /N for the le spacgs, where I ad N are two postve tegers. The crossg pots Ω τ are called the grd pots, h Ω τ h = x, t ) x =, =, 1,...,I; t = τ, =, 1,...,T /τ.

7 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Sppose U = I, s a grd fcto o Ω τ. For the smplcty, assme the costats a = b = 1. Itrodce h the followg otatos [4]: / = 1 ) /, = 1 ), δ x / = 1 h δ x = 1 h ), δt / = 1 ), τ 1 ), δ t = 1 τ δ x = 1 h 1 1 ), δ t = 1 ), 1 τ ). We deote the vale of the solto x, t) at the grd pot x, t ) by U follows from.14),.15),.1),.) that: ad f = f x, t ). Usg the Taylor expaso, t δt U δx U U = f R, 1 I 1, 1, 3.1) U = φ h), I, 3.) U 1 = U τ φ 1 h) S 1, I, 3.3) δ t U 1/ I δ x U / = δt U m I m=1 δ t U 1/ δ x U 1/ = δt U m m=1 m)τ m)τ If the solto x, t) s smooth eogh, there exsts a costat C, sch that R C h τ ), S 1 Cτ, P Ch τ ), Q Ch τ ). I Fτ s) ds P I,, 3.4) Fτ s) ds Q,. 3.5) Omttg the trcato errors 3.1) 3.5), we costrct a fte dfferece scheme of the redced problem.14).17): δt δx = f, 1 I 1, 1, 3.6) = φ h), I, 3.7) 1 = τ φ 1 h), I, 3.8) δ t 1/ I δ x / = δ t 1/ δ x 1/ = Fτ s) ds,, 3.9) δt m I m=1 m)τ δt m Fτ s) ds,. 3.1) m=1 m)τ Ths s a explct scheme wth global bodary codtos. I the followg, we cosder the stablty ad covergece of the scheme. Mltplyg 3.6) by hδt ad smmg p for from 1 to I 1, we obta =1 hδ t δ t δ x After some calclato, we get =1 hδ t δ t = h τ =1 ) = =1 δt 1/ hδ t f. 3.11) ) δt / ). 3.1) Usg the smmato by parts formla ad the bodary codto 3.9) 3.1), we have hδt δ x = h τ =1 = h τ = = δ x 1 1/ δ x 1/ h τ = δ x 1 1/ δ x 1/ h τ = δ x 1/ δ x 1/ δ x 1/ δ t δ x / δ t I δ x 1/ δ x 1/

8 1574 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) δt δ t 1/ δ t I I 1 I I 3. Notce that, I = δt δ t 1/ Smlarly, = δt δt I A IB. δ t 1/ I δt m m=1 m)τ δt m m=1 m)τ δt m m=1 m)τ m=1 δ t m I Fτ s) ds Fτ s) ds m)τ Fτ s) ds Fτ s) ds δt δt 1/ δt 3.13) I 3 I A 3 IB 3. Usg the lear terpolato to costrct a cotos fcto ũ1, t), whch satsfes ũ t 1, t) = δt m, t [mτ τ,mτ ). Accordg to the stablty estmate of the cotos case Secto, we obta I A = δ t δ t δt m m=1 m)τ It s easy to check, I B = δ t δt 1/ δt 1 = By the same techqe, we obta that I A 3 I B 3 = δ t I δt 1/ I δt I = 1 Fτ s) ds δt 1/ The thrd term the left of Eq. 3.11) s easy, =1 hδ t = h τ =1 1 h τ We trodce a axlary qatty Ẽ, Ẽ s h δ t 1/ = s h δ t 1/ =1 ) =1 ) =1 δt 1/ δt 1/. 3.14) ) δt / ). s also oegatve, ad δt 1/ I =1 1/) hτ δt 1/ 4 = s h δ t 1/ hs 4 = I αh δt 1/ = =1 ) 1 s τ ) h 4 δt 1 1/ δ t 1/ ) = ) δt / ) I ) ) s h δ t 1/ I ) s h δ t 1/ I =1 ) ) δx 1/ 1/ δt 1/ = δx 1/ 1/ ) =1 ) = ) = δ x 1 1/ δ x 1/ h δx 1/ 1/ ) s h δ t 1/ ) I s h 1/ ) =1 1/ =1 ) h 4 ) =1 = 1 δx 1 1/ δ x 1/ δt 1/ ) >, 3.16) ) )

9 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) where s = τ h, 1), α s = m, 1 s τ ) >. 4 The rght had of Eq. 3.11) s boded by: =1 hδ t f = h =1 f δt 1/ 1 s τ ) h 8 δ t / ) =1 δt / ) δt 1/ ) h 1 s τ /4 =1 f ). 3.17) Assme s τ 4 < 1, accordg to the defto of Ẽ, we obta the eqalty Ẽ 1 s τ ) h δt / ). 4 =1 3.18) Combg the expressos 3.1) 3.18), we get Ẽ Ẽ τ τ Ẽ Ẽ ) f, 1 s τ /4 Ẽ 1 τ / τ Ẽ f. 1 τ / 1 τ /)1 s τ /4) Whe τ 3,wehave Ẽ 1 3τ ) Ẽ τ f. 1 τ /)1 s τ /4) By dscrete Growall eqalty see page 11 of [3]), we arrve at Ẽ Ẽ 1 3τ ) τ f k1 1 τ /)1 s τ 1 3τ ) k /4) k= k=1 e 3 τ τ Ẽ f k. 1 τ /)1 s τ 3.19) /4) From 3.16) ad 3.19), we obta the stablty theory of the scheme 3.6) 3.1). k=1 Theorem 3.1 stablty of the scheme). Sppose s the solto of dfferece scheme 3.6) 3.1), s= τ /h, 1) ad α = m s, 1 s τ 4 )>.Let I E = αh δt 1/ ) δx 1/ 1/ = = ) =1 1/ deotes the eergy orm at th tme. We have the followg estmate, E Ẽ e 3 τ τ Ẽ 1 τ /)1 s τ /4) Where, f k = =1 h f k ). ) f k. k=1 Sce the dfferece scheme 3.6) 3.1) s a system of lear algebrac eqato at each tme level, t s easy to obta, Lemma 3.1. The dfferece scheme 3.6) 3.1) has a qe solto. Next, we tr to aalyze the covergece of the dfferece scheme. Let e = U x, t ). Sbtractg 3.1) 3.5) from 3.6) 3.1), we ca obta the error eqato: deotes the error o the grd pot

10 1576 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) δt e δx e e = r, 1 I 1, 1, 3.) e =, I, 3.1) e 1 = s 1, I, 3.) δ t e 1/ I δ x e / = δ t e 1/ δ x e 1/ = m=1 δ t em I m)τ δt em m=1 m)τ Fτ s) ds p I,, 3.3) Fτ s) ds q,. 3.4) Where there exsts a costat c, sch that r c h τ ), s 1 cτ, p I ch τ ), q ch τ ). 3.5) Usg the same techqe the proof of Theorem 3.1, we ca obta the followg covergece theory. Theorem 3. covergece of the scheme). Let E deotes the eergy orm of the error at th tme level, s = τ /h, 1),adα = s/. I E = αh δt 1/ = ) = δx 1/ 1/ we have the estmate, E e 3 τ τ k= q Ẽ ) 1 τ / αh p αh ad k=1 It follows from 3.1), 3.) ad 3.5) that, Ẽ = s h δ t e 1/ ) =1 = O Ihτ ) = O τ ), k= q ) αh p αh k=1 I ) =1 δt e 1/ ) I ) ) s h δ t e 1/ I hr ) O1). α =1 =1 1/ ) ) hr ). α = δ x e 1 1/ δ xe 1/ h Hece, the eergy orm of the absolte error E have oe order covergece. 4. The fast algorthm The artfcal bodary codtos eed to compte the covolto terms, whch are very expesve for mercal comptato. We recall the bodary codto.18) ad take a = b = 1, 1, t) 1, t) = Jt τ )1, τ ) dτ. Where the specal fcto Jt τ ) ca be defed by the seres: =1 e 1 e Jt τ ) = 1 J t τ ) J t τ ) = 1 ) l1 t τ )l l 1)!l )!l3 1 α l t τ ) l. l= l= 4.1) The coeffcets α l decay a rate of Ol! l ) ),sowejstchoosethefrst K 1K s a postve eve mber) terms 4.1) to approxmate the specal fcto Jt τ ). The we obta a approxmate bodary codto of the orgal bodary codto.18):

11 1, t) 1, t) = = K l= H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) K / K / l= l= P l t) α l t τ ) l α l l1) j= 1, τ ) dτ 1 C j l1) ) j t j τ l1) j 1, τ ) dτ 1, τ ) dτ 1 1, τ ) dτ τ l 1, τ ) dτ, 4.) here P l t), l =, 1,,...,K are gve polyomals of tme t. The advatage of ths algorthm s that we jst eed to deal wth a seres of tegratos τ l 1, τ ) dτ, l =, 1,,...,K, stead of the covolto term. Sce t τ l 1, τ ) dτ = τ l 1, τ ) dτ τ l 1, τ ) dτ, l =, 1,,...,K. t I practcal comptato, at the th tme level we jst eed to save the prevos tegrato vale, ad do oe step tegral calcls. Or mercal example shows that ths algorthm s very effcecy. 5. Dscrete artfcal bodary codtos I ths secto we dscss how to get the dscrete artfcal BCs DABC) for the Kle Gordo eqato. Ths approach was trodced by A. Arold [4] for Schrödger eqato, ad M. Ehrhardt [8] for parabolc eqato. After that Arold ad Ehrhardt sed ths approach to fd the dscrete artfcal BCs for other eqatos [5,11,5]. [5] deals wth a geeralzed Schrödger eqato appearg acostcs. [5] deals wth a parabolc eqato. [11] deals wth a Schrödger Posso system. A Prceto grop also adapted ths approach to systems of wave eqatos for materals wth cracks [7]. Istead of dscretzg the aalytc ABC lke 3.9) ad 3.1), we costrct DABCs of the flly dscretzed whole-space problem. Recosder the orgal tal vale problem.1),.), for smplcty assmg a = b = 1 aga. We mmc the dervato of the aalytc ABC Secto o a dscrete level. Frst choose two tegers I ad N, ad choose T as a fxed comptatoal tme. Let h deotes spatal mesh, τ deotes tme mesh, respectvely. h = 1/I, τ = T /N. Wth the form grd pots x, t ) x =, Z, t = τ, N ad the approxmatos x, t ) ad f f x, t ), the dscretzed Kle Gordo eqato o the boded doma, ) [, T ] reads: ) 1 α 1 ) τ = τ f, =, ±1, ±,..., = 1,,..., 5.1) = φ x ), 1 = τ φ 1 x ), =, ±1, ±,..., 5.) wth ) τ α =. h Assme ϕ x), ϕ 1 x), ad f x, t) have the same compact spport as Secto, we get =, 1 =, = I, I 1, I,..., =,,,..., f =, = I, I 1, I,..., =,,,..., =, 1,... We try to fd the bodary codto o Σ h I1 ad Σh, Σ h I1 = x I1, t ) =, 1,..., Σ h = x, t ) =, 1,.... Frst cosder the restrcto of the problem 5.1), 5.) for I, whch satsfes the followg dfferece eqato. 1 ) α 1 ) τ =, = I 1, I,..., = 1,,..., 5.3) = φ x ), 1 = τ φ 1 x ), = I 1, I, ) Sce I =, 3,... are kow o the bodary Γ h I1, the problem 5.3), 5.4) s complete, whch caot be solved depedetly. Assme I =, 3,... are gve, the the dfferece eqato above has a qe solto. Ths problem s defed o the half-fte doma. To solve t, we se the Z -trasform method see page 117 of [14]):

12 1578 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Let Z = U z) := z, z C, z > R, = where R deotes the covergece rads of ths Laret seres. Z s called the Z -trasform of the seqece for each fxed dex. Accordg to the tal codto ad defalt =, we have Z 1 = zu z), Z 1 = z U z). The dfferece eqato 5.3) becomes zu z) U z) 1 z U z) α U 1 z) U z) U z) ) τ U z) =, = I 1, I,... The U = I, I 1, I,..., satsfy the followg problem αu 1 βz)u z) αu =, = I 1, I,..., 5.5) U z),, 5.6) where βz) = z 1 z c, c = τ τ h ). Eq. 5.5) s a homogeeos d order dfferece eqato wth costat coeffcets, of whch the solto has the form: U z) = λz) ) I UI z), = I, I 1, I, ) The λz) satsfes αλ z) βz)λz) α =. 5.8) By Eq. 5.8) ad the assmptos 5.6), 5.7), we get: λz) = βz) β z) 4α = z 1/z c z r 1 z r r 3 /z 1/z α α = z 1/z c z 1 r 1 /z r /z r 3 /z 3 1/z 4 α z 1/z c zsz), 5.9) α where r 1 = r 3 = c, r = c 4α are three costats. Frst, we try to fd the Laret expaso of Sz). Observe that S z) satsfes, S z) = r 1/z r /z 3 3r 3 /z 4 4/z 5, Sz) hece S z) 1 r 1 z r z r 3 z 1 ) = r 1 3 z 4 z r z 3r 3 3 z 4 ) Sz). 4 z 5 5.1) Next, we assme that Sz) = 1 1 a z ad S z) = 1 a z, sg the formla 5.1) we ca obta a recrso relato of a for 5, wth a = 1 3 )r1 a 6 )r a 9 )r 3 a 3 1 )a 4, a 1 = r 1, a = r a 1r 1 4, a 3 = r 3 a r 1, a 4 = 1 a 1r 3 8 a r 4 5a 3r 1 8. Accordg to 5.9), we obta the Laret seres of λz) = = λ z : λ = c a 1 α, λ 1 = 1 a α, λ = a 1,. α From Fg. 1 we ca see that the absolte vale of λ decle very qckly. The method of comptg the Laret coeffcets throgh a ODE was frst gve Secto of Chapter 1 [1]. Now, we get the verse trasform of λz) Z λz) =λ.

13 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Fg. 1. The decay tedecy of the coeffcet λ,hereh =.1, τ =.5. Accordg to the formla 5.7) ad the covolto theorem for Z -trasforms, we get I1 = k= λ k k I, =, 3,... Ths s the DABC o the bodary Σ h I1, we are tryg to fd. Smlarly, we ca get the DABC o the bodary Σh as followg = k= λ k k, =, 3,... I practcal, we ca make some tables for the coeffcets λ before startg the mercal comptatos. Usg or DABCs, the mercal solto o the comptatoal doma D exactly eqals the restrcto of the dscrete whole-space solto o the comptatoal doma. Therefore, ths scheme prevets ay mercal reflectos at the bodary. Remark 5.1. Accordg to or calclato, these coeffcets λ are exact, the mercal error of the DABC method s jst eqal to the dscretzato error of 5.1), 5.) o the boded doma R 1 [, T ]. Hece the DABC method has the secod order covergece. Remark 5.. Oe ca also compte explct solto to homogeeos d order dfferece eqato wth costat coeffcets, see [9]. So wth oly very mor chages, we ca also deal wth the Kle Gordo eqato wth tal data that s ot spported wth the comptatoal doma. 6. Nmercal tests To show the effectveess of dfferet bodary codtos, ABC, DABC ad the fast algorthm FAST) are gve ths paper. We preset some mercal examples ths secto. I Example 1, we cosder the Kle Gordo eqato wthot sorce term, the exact solto s gve, ad the mercal soltos are compared wth the exact solto. The secod example s the Kle Gordo eqato wth sorce term, smltaeosly we compare the comptatoal tme of the dfferet schemes for Example. We also test the relato betwee the mercal accracy of the fast algorthm ad the optmal strategy of choosg K, especally, we test the log tme performace of the fast algorthm FAST). Example 1. We cosder the Kle Gordo eqato wthot sorce term: =, x R1, t,

14 158 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Table 1 Comptatoal errors ad covergece rate of the dfferece schemes for Example 1 t =.5 s). Mesh sze ABC FAST DABC 1/ 1.834E E.4531E 1/4 4.83E E E /8 1.3E E E /16.943E E E Table Comptatoal errors ad covergece rate of the dfferece schemes for Example 1 t = 1. s). Mesh sze ABC FAST DABC 1/ 8.834E E E 1/4.15E E E / E E E / E E E Table 3 Comptatoal errors ad covergece rate of the dfferece schemes for Example t = 1. s). Mesh sze ABC FAST DABC 1/ 8.77E E E 1/4.447E.9.659E E.4 1/ E E E / E E E s5π x), x 1, t= =, x > 1, t t= =, x R 1 whch has the exact solto: x, t) = 1 φ x t) φ x t) t xt x t φ ξ) J 1 t x ξ) ) t x ξ) dξ, where φ x) = t=. The solto represets two waves propagatg to the left ad rght respectvely wth ampltdes gradally decreasg. To evalate the qalty of mercal solto, we defe a error fcto as Et) = m, t) exa, t) L. exa, t) L The relatve error ad covergece rates of Example 1 are show Table 1 t =.5 s) ad Table t = 1. s).itcabe observed that the errors decay wth a early-optmal covergece rate of 4 whe the mesh s refed by a factor. Whe the comptato tme t =, all the orgal wave wll propagate ot the comptatoal doma. Example. Secodly, we cosder the same Kle Gordo eqato wth sorce term, whch wll physcally effect the wave propagatos. = 1 cos5t) s3π x), x R1, t, s5π x), x 1, t= =, x > 1, t t= =, x R 1. I ths example, the exact solto s gve o a very fe mesh h = 1 64, τ = h ). The relatve error ad covergece rates of Example are show Table 3 t = 1. s) ad Table 4 t = 1.5 s). It ca be observed that the errors decay wth a early-optmal covergece rate of 4 whe the mesh s refed by a factor. Fg. shows the wave ampltdes of dfferece schemes at fxed tmes, left oe s for Example 1 t = 1. s) ad rght oe s for Example t = 1.5 s). Compared wth the left oe, we ca fd that the exteral force ca geerate ew wave, whe the orgal wave propagate ot the comptatoal doma.

15 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Table 4 Comptatoal errors ad covergece rate of the dfferece schemes for Example t = 1.5 s). Mesh sze ABC FAST DABC 1/ E E.887E 1/ E E E /8 9.49E E E /16.187E E E 3.33 Table 5 Comptatoal tme of dfferet scheme for Example t = 1. s). h mesh sze) ABC DABC FAST 1/ / / / / / Table 6 The relato betwee comptatoal accracy ad K, for fxed mesh sze h = 1/3. Termal tme K Tme cost secods) Error T = 1 K = e 9 T = K = e 9 T = 3 K = e T = 4 K = e1 T = 5 K = e 9 T = 6 K = e T = 7 K = e 9 T = 8 K = e T = 9 K = e 9 T = 1 K = e 9 T = 1 K = e 9 T = 16 K = e 9 T = K = e 9 Fg.. Wave ampltdes of dfferet schemes at fxed tmes, left s for Example 1 t = 1. s),rghtsforexamplet = 1.5 s). Table 5 shows comptatoal tme of dfferece scheme for Example t = 1. s). The ABC method s very expesve for mercal comptato, whe the mesh s very fe. The FAST algorthm mproves the effcecy dramatcally, ad the DABC method s very fast too. Table 6 shows the relato betwee comptatoal accracy of the fast algorthm ad the optmal strategy of choosg K, here the Error fcto s defed by ErrorT ) = m, T ) exa, T ) L. I Eq. 4.) of Secto 4, we trcate the power seres expaso of the specal fcto Jt τ ) to obta a fast algorthm.

16 158 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Fg. 3. The relatoshp betwee the choosg of K ad the accracy of the reslt for Example wth the fxed tme t = 5, h = 1/16. Fg. 4. The decay rate of the error L orm choosg dfferet K for Example wth the fxed tme t = 5. From left to rght, K =,, 4, 6, 8, 1, h = 1/16. Nmercal tests dcate that the mercal accracy of the fast algorthm hghly depeds o the comptatoal tme t ad the choosg of trcato term mber K. Whe the comptatoal tme s ot very log compared wth the spatal doma [, 1], oly sg few terms wth K small) ca get a very hgh accracy. For log tme comptato, order to get the same accracy, we mst crease the term mber K accordgly. Fg. 3 shows the relato betwee comptatoal accracy ad the dfferet mber K. For the fxed tme t = 5, we ca see that whe K 4, the mercal solto s almost the same wth the exact solto. Fg. 4 shows the covergece rate betwee the mercal solto ad the exact solto for dfferet K. Fg. 5 shows the log tme propagato of the Kle Gordo eqato. 7. Coclso I ths paper, we aalyze the fte dfferece method for the oe-dmesoal Kle Gordo eqato o the boded doma. Two artfcal bodary codtos are obtaed to redce the orgal problem to a tal bodary vale problem o a boded comptatoal doma, whch s dscretzed by a explct dfferece scheme. The stablty ad covergece of the scheme are aalyzed by the eergy method. A fast algorthm s obtaed to redce the comptatoal cost ad a

17 H. Ha, Z. Zhag / Appled Nmercal Mathematcs 59 9) Fg. 5. Log tme t = s) comptato of Example. dscrete artfcal bodary codto DABC) s derved by the Z -trasform approach. Fally, we llstrate the effcecy of the proposed method by several mercal examples. The artfcal bodary codto for the mlt-dmesoal ad olear Kle Gordo eqato wll be cosdered as or frther work. Refereces [1] L.C. Adrews, Specal Fctos of Mathematcs for Egeers, McGraw-Hll Ic., New York, 199. [] X. Atoe, C. Besse, Ucodtoally stable dscretzato schemes of o-reflectg bodary codtos for the oe-dmesoal Schrödger eqato, J. Compt. Phys ) [3] X. Atoe, C. Besse, V. Moysset, Nmercal Schemes for the smlato of the two-dmesoal Schrödger eqato sg o-reflectg bodary codtos, Math. Compt ) 4) [4] A. Arold, Nmercally absorbg bodary codtos for qatm evolto eqatos, VLSI Desg ) [5] A. Arold, M. Ehrhardt, Dscrete trasparet bodary codtos for wde agle parabolc eqatos derwater acostcs, J. Compt. Phys ) [6] D.B. Dca, Symplectc fte dfferece approxmatos of the olear Kle Gordo eqato, SIAM J. Nmer. Aal ) [7] W. E, Z.Y. Hag, A dyamc atomstc-cotm method for the smlato of crystalle materals, J. Compt. Phys. 18 ) [8] M. Ehrhardt, Dscrete trasparet bodary codtos for parabolc eqatos, : Proceedgs of the GAMM 96 Coferece, ZAMM ) [9] M. Ehrhardt, Dscrete trasparet bodary codtos for Schrödger-type eqatos for o-compactly spported tal data, Appl. Nmer. Math. 58 8) [1] M. Ehrhardt, Dscrete artfcal bodary codtos, Ph.D. Thess, TU Berl, 1. [11] M. Ehrhardt, A. Zsowsky, Fast calclato of eergy ad mass preservg soltos of Schrödger Posso systems o boded domas, J. Compt. Appl. Math ) 1 8. [1] B. Egqst, A. Majda, Absorbg bodary codtos for the mercal smlato of waves, Math. Compt ) [13] D. Gvol, Hgh-order oreflectg bodary codtos wthot hgh-order dervatves, J. Compt. Phys. 17 1) [14] I.S. Gradshtey, M. Ryzhk, Table of Itegrals, Seres ad Prodcts, Academc Press, New York, 4. [15] W. Greer, Relatvstc Qatm Mechacs-Wave Eqatos, thrd ed., Sprger, Berl,. [16] H. Ha, Z. Hag, Exact artfcal bodary codtos for the Schrödger eqatos R, Comm. Math. Sc. 4) [17] H. Ha, D. Y, Absorbg bodary codtos for the mlt-dmesoal Kle Gordo eqato, Comm. Math. Sc. 5 7) [18] H. Ha, D. Y, Z. Hag, Nmercal soltos of Schrödger eqatos R 3, Nmer. Methods Partal Dfferetal Eqatos 3 7) [19] H. Ha, C.X. Zheg, Exact oreflectg bodary codtos for exteror problems of the hyperbolc eqato, Chese J. Compt. Phys. 5) [] R.L. Hgdo, Ratoal bodary codto for dspersve waves, SIAM J. Nmer. Aal ) [1] R.L. Hgdo, Absorbg bodary codtos for dfferece approxmato to the mltdmesoal wave eqato, Math. Compt ) [] M.E. Khalfa, M. Elgamal, A mercal solto to Kle Gordo eqato wth Drchlet bodary codto, Appl. Math. Compt. 16 5) [3] B.G. Pachpatte, Ieqaltes for Fte Dfferece Eqatos, Moographs ad Textbooks Pre ad Appled Mathematcs, Marcel Dekker Ic., New York,. [4] X.N. W, Z.Z. S, Covergece of dfferece scheme for heat eqato boded domas sg artfcal bodary codtos, Appl. Nmer. Math. 5 4) [5] A. Zsowsky, M. Ehrhardt, Dscrete trasparet bodary codtos for parabolc systems, Math. Compt. Model. 43 6)