NONLINEAR SCHRÖDINGER EQUATION

Transcription

1 TMA Nuerical Soluio of Differeial Equaios by Differece Mehods NONLINEAR SCHRÖDINGER EQUATION Group Cadidaes 8, 37 ad 55 Depare of Maheaical Scieces Norwegia Uiversiy of Sciece ad Techology April, 3 Absrac I his projec we cosider four fiie differece schees for he periodic oe-diesioal cubic oliear Schrödiger equaio: The Crak Nicolso ehod, a liearized Crak-Nicolso ehod, a eplici Dufor-Frakel ype ehod, ad a las a relaaio schee by Besse. Resuls fro several uerical eperies are give, ad we ivesigae a coserved quaiy. Coes. Iroducio. Discreizaio 3.. Crak Nicolso ehod Liearized Crak Nicolso ehod Du For-Frakel ype ehod Besse relaaio schee Sabiliy aalysis for he Crak-Nicolso ehod. Ipleeaio ad opiizaio i MATLAB 7 5. Nuerical eperies Eaple Eaple Eaple A coserved quaiy Coclusio Refereces of

2 of Noliear Schrödiger Equaio. Iroducio The oe-diesioal cubic oliear Schrödiger equaio reads { i u + u = λ u u, u, ) = f),, ) T R, ) where u is a cople-valued fucio ad λ R is a cosa. The circle group T, which is isoorphic o R/pZ, defies periodic boudary codiios wih period p, ad aily we choose o use p = ad work o he ierval [ π, π] Re u Figure : The real par of a plae wave soluio wih paraeers λ =, A =, B = ad C = o he ie ierval [, ]. 3 3 Equaio ) arises frequely i aheaical physics, ad occurs for eaple i he odelig of deep waer waves i hydrodyaics ad he propagaio of pulses i oliear opics. The paraeer λ describes eiher a focusig λ < ) or defocusig λ > ) effec of he olieariy. I is show i [3] ha here eiss a ifiie uber of soluios of ), ad i fac his resul holds for ay give period. Furherore, oe ca esablish ha ) has aalyic plae wave soluios of he for u, ) = A ep [ i B A λ + B ) + C )], ) where A, C R ad B Z are arbirary cosas, ad Figure shows a eaple of hese. u Figure : The absolue value of 3) o he iervals [ 3π, 3π] ad [ 5, 5].

3 Group Cadidaes 8, 37 ad 55 3 of More ieresigly are he well-kow solio soluios of he oliear cubic Schrödiger equaio, bu we have o bee able o fid ay lieraure o eplici soluios for he geeral periodic case. However, [] eios he bi-solio soluio i + i) cosh sih u, ) = e cosh) 3) for he case λ =, which igh be cobied o give a spaial periodic soluio. To obai his, oe has o glue ogeher ay copies of he soluio show i Figure. Furherore, i is kow ha ) has a ifiie uber of ivarias, ad especially oe ca show ha he ass M[u] = is a coserved quaiy. T u, ) d ). Discreizaio To begi discreizaio, we iroduce a recagular esh wih odes, ) = π + h, k) for =,..., M + ad Z, where he spaial ad eporal sepsizes are h = π/m + ) ad k, respecively. We deoe by u he eac soluio of ) a he esh poi, ), wih U correspodigly he uerical approiaio o u, ad U = U, U,..., U M ). ha by he periodiciy of he proble U = U M+, ad hece we skip his copoe i U. Noe May fiie differece schees of ) have bee developed, ad we hough i ieresigly o iplee ad copare soe of he. Firs we will sar by lookig a he Crak-Nicolso schee, ad e obai a liearized versio by reaig he oliear er as i []. Aferwards, he so-called Du For-Frakel ype eplici ehod is preseed, ad a las we cosider he Besse relaaio schee... Crak Nicolso ehod Le u + = u, + k). By wriig ) i he for u = i u iλ u u, we obai by usig he rapezoidal rule i ie ad ceral differeces i space ha = u + k u + u + ) k3 3 u + + Ok ) = u + k [ i u iλ u u ) + i u + iλ u + u + )] k3 3 u + + Ok ) = u + [ ik h δ u + ) h δ u + h u + )] h u + + Oh ) ikλ u u + u + u + ) k3 3 u + + Ok ). u + Seig r = k/h we ed up wih u + = u + ir δ u + δu + ) ikλ u u + u + τ = k3 3 u + ikh u + + Ok + k 3 h + kh ), u + ) + τ, 5)

4 of Noliear Schrödiger Equaio where i is used ha u + u + ) = u + + Ok ). Disregardig τ i 5) gives he geeral Crak-Nicolso ehod: or wrie ou: U + = U + ir δ U + δu + ) ikλ U U + U + U + ), ru i r)u + ru+ + = ru + i + r)u ru+ + kλ U U + U + U + ) ). This resuls i a oliear syse A + U + = A U + kλ U +. U + + U. U ), 7) where ± i r ) ± i r ) A ± = ±r ± i r ) is a syeric cyclic ridiagoal ari of diesio M + ) M + ). Moreover, we le deoe copoewise vecor uliplicaio, ad siilarly defie U +. = U +,..., U + M ). The local rucaio error is τ = k τ = k 3 u + ih u + + Ok 3 + k h + h ) = Ok + h ), ad hus he ehod is cosise. Is sabiliy is show i Secio 3... Liearized Crak Nicolso ehod As suggesed i [], a liearized versio of he Crak-Nicolso schee ca be foud by epadig he las oliear er i 5) as follows: u + u + = u u + k u u ) + Ok ) = u u + k u u u + k u u + Ok ). I order o avoid ay + )h ie-level ers o he righ-had side, i is aural o approiae he eporal derivaive usig a backward differece: u = k Hece we ed up wih u u ) + Ok) ad u = k u + u + u u ) + Ok). = u u u u u u u + Ok ).

5 Group Cadidaes 8, 37 ad 55 5 of Iserig his io 5), wriig ou he spaial ceral differece operaors ad eglecig he local rucaio error gives a liearized versio of he Crak-Nicolso ehod: or ru i r)u + ru+ + = ru + i + r)u ru+ + kλ 5 U U U U U U U ), A + U + = A U + kλ 5 U. U U U U U. U ), 8) siilarly as i 7). The soluio of he firs ie-layer is foud via he ordiary Crak-Nicolso schee. Furherore, afer suig up he ers we have τ = Ok + h ) as before..3. Du For-Frakel ype ehod Aoher schee is he so-called hree-level eplici DuFor-Frakel ehod. As eplaied i [], he discreizaio of ) reads i U + U k = U + U + U + U h + λ U U + + U, which leads o U + = + ir + kλ U ) ) [ U ir + kλ U ) ) + ir U+ + U )], 9) where r = k/h. Siilarly as for he liearized Crak-Nicolso schee, he soluio of he iiial eporal sep is foud via he ordiary Crak-Nicolso schee. Oe ca show [5] ha he local rucaio error is of order Ok + h + k/h) ), ad hece we eed k/h for cosisecy... Besse relaaio schee The las ehod we have sudied is kow as he Besse relaaio schee []. Is key idea is o replace ) wih a syse of wo equaios { v = u, i u + u = λvu,, ) T R, ) wih u, ) = f) as before. Now defie + v a he poi, + = + k ad le V + be a approiaio of ). The Besse suggess he followig fiie differece schee for ): V + + V = U, i U + U + U + k h δ + U ) which ca wrie as he coupled syse U + = λ + U ) V +, { V + = V + U., A + U + = A U, )

6 of Noliear Schrödiger Equaio wih V ± = V ±,..., V ± ). Here M+ α ± α ± A ± = ±r α ± M ) is a syeric cyclic ridiagoal ari, α ± = ±i/r) λh V + ad r = k/h. Furherore, Besse proposes o se U = f ),..., f M+ )) ad V = U. for he firs eporal sep. Our discreizaio gives order wo i space, ad he schee is prove o be coverge i [], wih idicaios ha he order of covergece equals wo i ie. A advaage wih he Besse relaaio ehod is ha we do o have o solve a oliear syse. 3. Sabiliy aalysis for he Crak-Nicolso ehod We ow show sabiliy for he Crak-Nicolso schee usig a vo Neua aalysis echique. Firs, for oaio, defie U a ) { = a U, U + }. 3) =,...,M+ Furherore, le U = ξ e ihβ, wih ξ >. Fro ) we he have ξ + e ihβ re ihβ + i r) + re ihβ) ξ e ihβ re ihβ i + r) + re ihβ) + kλ ξ e ihβ ξ e ihβ + ξ kλ + e ihβ ξ + e ihβ ξ e ihβ re ihβ i + r) + re ihβ) kλu aξ ) e ihβ kλu + ) aξ + e ihβ. ) Afer dividig by ξ e ihβ ad rearragig he ers we obai ξ i r + r cosβh) ku aλ ) ) By a riagle iequaliy a b a b, we also have ha i r + r cosβh) ku aλ ) ) i r + r cosβh) ku ) aλ i r + r cosβh) ku ) aλ. 5). ) Cobiig 5) ad ) we see ha i order for he boh o hold we eed ξ. Thus, fro he vo Neua s sabiliy crierio we have prove ha he Crak-Nicolso schee is ucodiioally sable for he oliear Schrödiger equaio. Sice he ehod also is cosise, La equivalece heore saes ha he ehod is coverge. Hece, we have prove ha he Crak-Nicolso schee is boh coverge ad sable for he oliear Schrödiger equaio.

7 Group Cadidaes 8, 37 ad 55 7 of. Ipleeaio ad opiizaio i MATLAB The four differece schees are divided io separae fucios i MATLAB, ad he called fro a ai scrip. The ipleeaios ca hadle differe equaio paraeers ad grid sizes effecively, ad hus ca be applied o siilar probles as well. I geeral, he uerical soluio is sored as a ari, wih rows correspodig o eporal soluios. Whe he syse of equaios is big, he values of he soluios are preferably wrie o a file as hey are copued. I his way oe does o have o sore he values a each sep, ad will herefore be able o save a lo of eory. We have o doe his i our case. Sice our syse is o ha big, we do o cosider i o be appropriae. To sore all he values of he soluios as hey are copued also akes he ploig uch easier. Moreover, by he periodiciy of he proble, we drop he las spaial copoe o avoid uecessary copuaios. Whe i coes o he Besse relaaio schee, he Crak-Nicolso ehod ad is liearized versio, he arices A ± ad A ± are ipleeed i sparse ridiagoal for usig he efficie coad gallery'ridiag') wih suiable paraeers. For he Besse relaaio schee ), we updae oly he diagoals of he arices A ± oce every ie sep i order o use less resources. To solve he oliear syse 7) for he Crak-Nicolso schee, we firs ipleeed a siple versio usig MATLABs fsolve fucio. However, his is slow ad resrics he grid size heavily, ad we isead develop a Newo-Raphso ehod as follows: Fro 7) defie F U + ) = A + U + A U kλ U +. U + + U. U ). We he wa o solve F U + ) =. The Jacobia of F equals DF U + ) = A + kλ diag U + U + + U +. ), ad he Newo-Raphso ieraio schee akes he for ) DF U + j) ) U + j+) U + j) = F U + j) ). 7) Sice boh he Jacobia of F ad he arices A ± i he Besse relaaio schee are syeric cyclic ridiagoal, we could have used a facorizaio echique, called Tepero s algorih, as described i [8]. The idea is o obai he firs ukow value U, ad he ge a siplified ridiagoal syse which ca be solved efficiely wih ehods like he Thoas algorih [9] or a LU facorizaio. By usig his ehod, he uber of operaios are epeced o decrease fro OM 3 ) o OM). The las par of DF is ipleeed wih spdiags. As a soppig crierio for 7), we check wheher U + j+) U + j) is less ha a give olerace, ad he sarig poi is chose o be he soluio of he previous eporal sep. We use a olerace equal o 5. This is very close o he achie epsilo, ad fro uerical eperies, we observe ha avoidig a olerace equal o he achie precisio gives covergece i orally less ha ieraios. A las, wih he ecepio of he liearized Crak-Nicolso schee which eed o updae i is arices, o iverses are copued, ad we use he fas rdivide fucio for icreased speed. Furherore, we precopue cosas ad avoid repeaed evaluaios where possible.

8 8 of Noliear Schrödiger Equaio 5. Nuerical eperies I his secio we prese hree uerical eaples for he oliear Schrödiger equaio ) wih esiaio of ruies ad soe error aalysis. Moreover, he coservaio of a discree aalog of ) is cosidered. 5.. Eaple We ow look a he ierval [, 5] Schee M N io f ) = ep i) is chose, which Crak-Nicolso correspods o he aalyical solu- Liearized Crak-Nicolso.33 io give i ) wih λ =, A =, Du For-Frakel 5.3 B = ad C =. A copariso Besse relaaio.33 ad [ π, π]. The iiial fuc- of he ruie for our four differe Ruie Table : Copariso of ruie for four schees for eaple. Ruies are give i secods. schees is give Table. I order o aalyze he ruies for he differ- e schees we choose equal grid sizes for hree of our ehods. The Du For-Frakel ehod requires k h, ad hece we eed a fier grid o esure a siilar error. As epeced, he Crak-Nicolso schee is by far he slowes ehod, sice i requires solvig a oliear syse a each sage. Moreover we observe ha he liearized Crak-Nicolso versio ad he Besse relaaio schee ru a alos equal speed. A las we oe ha alhough he Du For-Frakel ehod is oly codiioally sable, our ipleeaio is quie efficie ad ca hadle large values for N wihou probles. Crak-Nicolso Du For-Frakel Liearized Crak-Nicolso 3 8 Besse relaaio Figure 3: Plos showig he absolue error bewee he aalyic -ad uerical soluio i Eaple for he four differece schees. a each poi i he grid. ChagLe he absolue error be defied as e = u, ) U ig M ad N, we ca see how e evolves. I Figure 3 is show he absolue error bewee he aalyic -ad uerical soluio for he four differece schees wih grid sizes as i Table. All ehods display a very siilar behaviour, bu he Crak-Nicolso -ad Besse relaaio schees give he lowes global error. For speed he, he Besse relaaio schee ay be he preferred oe.

9 Group Cadidaes 8, 37 ad 55 9 of Crak-Nicolso Li. Crak-Nicolso Du For-Frakel Besse relaaio Oh ) Crak-Nicolso Li. Crak-Nicolso Besse relaaio Ok ) ) log u,n) U N u,n) ) 3 log u, ) U u, ) h 3 k Figure : Verificaio of order of he error for he four differece schees usig log-log plos. Furherore, we also wa o verify he order of he local rucaio error τ for he differe schees. For verificaio of spaial order, we use a fie grid i ie wih N = 5 for boh he Crak-Nicolso schees ad he Besse relaaio ehod, bu N = for he Du For-Frakel ype algorih. We plo he relaive error u, N ) U N ) log u, N ) agais h for he las ie sep N, where u, N ) deoes a vecor coaiig he eac soluio i space evaluaed i he grid pois a ie N. Siilarly we check he eporal order wih M = 5 for all he differece schees, wih he ecepio of he Du For-Frakel ehod. I is possible o verify his also for his oe by eliiaig spaial order. However, his is beyod he scope of his projec. The relaive error ) u, ) U log u, ) is ploed agais k, where U deoes he uerical soluio i ie a =. I Figure he resuls are show usig he aalyic soluio eioed earlier i Eaple. The differe lies are shifed soewha i order o ge a beer feelig for he slopes. As we ca see, all ehods see o achieve secod order i boh space ad ie, wih he ecepio of he Du For-Frakel ype ehod. I space he order is, bu eve hough we use a very sall eporal sep size, he spaial sep size eveually caches up. Thus, sice i is oly codiioally sable, we see he curvaure a he ed. Nuerical ess have show ha if λ is oo big, he waves will be oo close ad our uerical rouie will require a ereely fie grid. We eperieced ha he Besse relaaio schee perfored slighly beer ha he oher schees, bu i will also "fail" if λ is big eough. The oher schees will eveually have waves shifed fro he aalyical soluio. We eperieced ha he focusig case λ < i geeral resuled i larger ru ies.

10 of Noliear Schrödiger Equaio 5.. Eaple We ow choose he iiial fucio f ) = si) wih [, 5], [ π, π] ad λ =. I figure 5 is show boh he real ad iagiary par of he uerical soluio usig Besse relaaio schee usig M = 5 ad N =. Ieresigly, he soluio is also periodic i ie, ad oreover, he real ad iagiary par of he soluio are equal ecep for a eporal raslaio. I u Re u Figure 5: Real ad iagiary par of he uerical soluio wih f ) = si) ad λ = Eaple 3 A las we cosider he bi-solio soluio give i 3). For siplic- iy, we have oly looked a he isolaed case, ha is, spaial peri- odic gluig of he soluio is oi ed. I order o iclude all of he bi-solio, i is ecessary o use a 5 larger spaial grid. I Figure is ploed he uerical error usig he Besse rela aio schee wih M = ad 5 N =. We observe ha he er- ror is larges a he posiio of he 5 5 Figure : The absolue error for he bi-solio for [ 5, 5] ad [ π, π] usig Besse relaaio schee. bi-solio.

11 Group Cadidaes 8, 37 ad 55 of 5.. A coserved quaiy I ca be show [, ] ha boh he Crak-Nicolso -ad he Besse relaaio schee coserves he discree varia M+ M[U ] = h U 8) of he quaiy i ). Le he spaial -ad eporal ierval be [ π, π] ad [, 5], respecively. The M[u] = π for he aalyic plae wave soluio ) wih paraeers λ =, A =, B = ad C =. I Figure 7 is ploed he differece bewee he aalyic ad discree versio 8) for he four differe ehods. = Crak-Nicolso 5 Li. Crak-Nicolso M[u] M[U ] M[u] M[U ] Du For-Frakel Besse relaaio M[u] M[U ] M[u] M[U ] Figure 7: Copariso of he coserved quaiy for he four differece schees. As we ca see, boh he Crak-Nicolso ehod ad he Besse relaaio schee see o preserve he ass correcly. The error is close o achie epsilo. Moreover, i looks like he liearized Crak-Nicolso ehod very slowly icreases M[U ] as ie goes. The Du For- Frakel ype ehod is resriced o a shor ie ierval i order o display is oscillaory behaviour. However he oscillaios are alos cosa i ie, so he ass ay be cosidered o be preserved also for he Du For-Frakel ehod.

12 of Noliear Schrödiger Equaio. Coclusio We have proved he ucodiioal sabiliy of he Crak-Nicolso schee ad is covergece for he periodic oliear cubic Schödiger equaio. Moreover, uerical ess verify ha he four fiie differece schees are of secod order i ie ad space wih he ecepio of Du For-Frakel ype ehod, which we have oly show o have order wo i space. Moreover, boh he Crak-Nicolso schee ad he Besse relaaio ehod are uerically verified o preserve he coserved quaiy ass. Our uerical eperies idicaes ha he Besse relaaio schee ay be cosidered as he bes ehod sice i is boh fas ad accurae, while he Crak-Nicolso schee see o provide beer sabiliy for large ie iervals. I pracice, if λ is big eough, he waves will be so close ad our uerical ehods will require a very fie grid o approiae he soluio well. Refereces [] Chrisophe Besse, A relaaio schee for he oliear schrödiger equaio, SIAM J. Nuer. Aal. ), o. 3, [] A.G. Brasos, A liearized fiie-differece ehod for he soluio of he oliear cubic schrödiger equaio, Korea Joural of Copuaioal ad Applied Maheaics 8 ), 59 7 Eglish). [3] L. Brüll ad H.-J. Kapelle, Periodic soluios of oe-diesioal oliear schrödiger equaios, Aca Maheaica Hugarica 5 989), o. 3-, 9 95 Eglish). [] More Dahlby ad Bryjulf Owre, Plae wave sabiliy of soe coservaive schees for he cubic schrödiger equaio, ESAIM: Maheaical Modellig ad Nuerical Aalysis 3 9), [5] F. Ivaauskas ad M. Radziuas, Sabiliy ad covergece of dufor-frakel-ype differece schees for a oliear schrödiger-ype equaio, Lihuaia Maheaical Joural ), 9 3 Eglish). [] D. H. Peregrie, Waer waves, oliear schrödiger equaios ad heir soluios, The ANZIAM Joural 5 983), 3. [7] Adrei D. Polyai, Schrodiger equaio wih a cubic olieariy, hp://eqworld.ipe.ru/e/ soluios/pde/pde.pdf,, [Olie; accessed -March-3]. [8] Clive Tepero, Algorihs for he soluio of cyclic ridiagoal syses, Joural of Copuaioal Physics 9 975), o. 3, [9] Wikipedia, he free ecyclopedia, Tridiagoal ari algorih, hp://e.wikipedia.org/wiki/ Tridiagoal_ari_algorih, 3, [Olie; accessed -April-3]. [] Lii Wu, Dufor frakel-ype ehods for liear ad oliear schrödiger equaios, SIAM J. Nuer. Aal ), o., [] Adrea Zisowsky ad Mahias Ehrhard, Discree arificial boudary codiios for oliear schrödiger equaios, Mah. Copu. Model. 7 8), o. -, 83.