Research Article An Optimized Runge-Kutta Method for the Numerical Solution of the Radial Schrödinger Equation

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1 Mathematcal Problems n Engneerng Volume 212, Artcle ID , 12 pages do:1.1155/212/ Research Artcle An Optmzed Runge-Kutta Method for the Numercal Soluton of the Radal Schrödnger Equaton Qnghe Mng, Yanpng Yang, and Yongle Fang School of Mathematcs and Statstcs, Zaozhuang Unversty, Zaozhuang 27716, Chna Correspondence should be addressed to Yongle Fang, ylfangmath@gmal.com Receved 2 July 212; Accepted August 212 Academc Edtor: Gradmr Mlovanovc Copyrght q 212 Qnghe Mng et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. An optmzed explct modfed Runge-Kutta RK method for the numercal ntegraton of the radal Schrödnger equaton s presented n ths paper. Ths method has frequency-dependng coeffcents wth vanshng dsperson, dsspaton, and the frst dervatve of dsperson. Stablty and phase analyss of the new method are examned. The numercal results n the ntegraton of the radal Schrödnger equaton wth the Woods-Saxon potental are reported to show the hgh effcency of the new method. 1. Introducton In ths paper, we are concerned wth the numercal ntegraton of the one-dmensonal Schrödnger equaton of the form y x v x E y x, 1.1 where the real number E s the energy and the functon v x s the effectve potental satsfyng v x asx. Two boundary condtons are assocated wth ths equaton: one s y, and the other mposed at large x s determned by physcal consderatons. The form of ths second boundary condton depends crucally on the sgn of the energy E. Such problems are frequently encountered n a varety of scentfc felds and engneerng applcatons 1 9. Concernng the oscllatory character of the soluton to the Schrödnger equaton 1.1, there have appeared a lot of numercal ntegrators of adapted type, a pronounced class of whch s based on mportant propertes such as the phase lag and the amplfcaton see These are actually two dfferent knds of truncaton errors.

2 2 Mathematcal Problems n Engneerng The frst s the angle between the analytcal soluton and the numercal soluton, and the second s the dstance from a standard cyclc soluton. If a good frequency s estmated n advance, then t s a good choce to construct numercal methods wth zero dsperson or/and zero dsspaton. These technques are called phase ftted or/and zero dsspaton. Related work can be founded n For Runge-Kutta methods, Smos and Aguar 18 constructed a modfed Runge-Kutta method for the numercal ntegraton of the Schrödnger equaton by phase fttng based on the ffth-order RK method. Recently, Van de Vyver 16 gave an embedded par of modfed RK methods by nullfyng the phase-lags of the ffthorder method and the fourth-order method. And n 22, Tstouras and Smos constructed phase-ftted and zero dsspaton ffth-order Runge-Kutta method for the numercal soluton of oscllatory problems. In ths paper, nspred by the deas n 2 28, we construct a new knd of modfed ffth-order Runge-Kutta method by nullfyng the dsperson, the dsspaton, and the frst dervatve of the dsperson. In Secton 2, the prelmnares of the phase propertes of explct modfed Runge-Kutta methods are ntroduced. In Secton, the coeffcents of a new knd of optmzed modfed RK method are obtaned. Secton 4 examnes the stablty and phase propertes of the new method. In Secton 5, the numercal experments are reported. 2. Prelmnares We begn by consderng the numercal ntegraton of the ntal value problem IVP of frstorder dfferental equatons n the followng form: y x f x, y ), y x y, 2.1 whose soluton shares an oscllatory character. We follow the conventon to assume that the frequency s known to be ω n advance or can be accurately estmated. An s-stage-modfed explct Runge-Kutta RK method has the followng scheme: 1 Y γ y n h a j f ) x n c j h, Y j, j 1 y n 1 y n h s b f x n c h, Y, 1 1,...,s, 2.2 where the coeffcents a j, c, b, 1,...,sare constants, h s the step sze, and the parameters γ, 1,...,s are even functons of ν hω. It s convenent to express the modfed RK method 2.2 by the Butcher tableau as follows: c 1 γ 1 c 2 γ 2 a c s γ s a s1 a ss 1 2. b 1 b s 1 b s or smply by c, γ, A, b. The extra-frequency-dependng parameters γ ν, ν hω, 1,...,s are ntroduced to tune the tradtonal RK method to the specal oscllatory structure of

3 Mathematcal Problems n Engneerng the problem. We assume that lm ν γ ν 1, 1,...,s so that as ν, the modfed RK method 2.2 reduces to a tradtonal RK method. An alternatve approach adopted by, for example, exponental/trgonometrc fttng technques, s to let some of the coeffcents a j, c, b, 1,...,sbe functons of ν hω see 16, 18, 29. Applyng the modfed RK method 2.2 to the test equaton as follows: y ωy, ω > 2.4 yelds y n 1 R ν y n, ν ωh. 2.5 A comparson of the numercal soluton wth the exact soluton leads to the notons of phaselag and dsspaton error defned as follows. Defnton 2.1. The followng two quanttes are called the phase lag or dsperson and the amplfcaton factor error or dsspaton error, respectvely: P ν ν arg R ν, D ν 1 R ν. 2.6 The method s sad to be dspersve of order q and dsspatve of order p f P ν O ν q 1), D ν O ν p 1). 2.7 If P ν andd ν, the method s called phase ftted zero dspersve) and amplfcatonftted zero dsspatve), respectvely. For modfed RK method 2.2, we have R ν U ν 2) νv ν 2), 2.8 where U ν 2) 1 t 2 ν 2 t 4 ν 4, V ν 2) 1 t ν 2 t 5 ν are polynomals n ν 2, whch are completely defned by the Runge-Kutta coeffcents c, A, γ, and b. Therefore, we have P ν ν arctan ν V ν 2) ), D ν 1 U ν 2 U ν 2 2 ν 2 V ν Based on the ffth algebrac order sx-stage Dormand and Prnce Runge-Kutta method, Smos and Aguar 18 obtaned an explct modfed RK method wth one parameter γ 2 takng the orthers γ 1 γ 1for,...,6 determned by nullfyng the quantty

4 4 Mathematcal Problems n Engneerng tan ν ν V ν 2 / U ν 2. In 22, Tstouras and Smos presented an optmzed Runge- Kutta method by nullfyng the dsperson and the dsspaton. In ths paper, we construct a new optmzed Runge-Kutta method by nullfyng the dsperson, the dsspaton, and the frst dervatve of the dsperson.. Constructon of the New Method In ths secton, we are concerned wth the followng Runge-Kutta method gven by the Butcher tableau as follows: γ γ γ If we choose γ 2 γ γ 4 1, the classcal Runge-Kutta method wth order ffth derved by Dormand and Prnce s recovered. In order to construct the new embedded RK par, we set γ 2,γ,γ 4 free and keep the rest of the coeffcents. Motvated by the deas n 2 28, we obtan the dsperson, the dsspaton, and the frst dervatve of the dsperson of ths method, whch depend on ν, γ 2,γ,γ 4 as follows: P ν tan ν M N, d ν 1 M 2 N 2,.2 where der P ν sec 2 ν M N MN N 2, M 15ν γ ν γ 2 ν 4 ) )) 111γ 4 2ν γ 71ν 2 15, N ) γ 265γ 4 729γ 5 ν γ 2 8γ 4 ) ν 4 762ν 6)..

5 Mathematcal Problems n Engneerng 5 Now, solvng.2, wegetγ values n terms of ν. Instead of gvng the very complcated expressons for γ, for the purpose of practcal computaton, we present ther Taylor expansons as follows: γ ν ν ν ν8, γ 1 91ν ν ν ν1, γ ν ν ν ν In order to check the algebrac order of the newly obtaned modfed RK method, we note that the order condtons lsted n 1 for tradtonal RK methods are not suffcent for the modfed RK method 2.2. Wrtng γ 1 γ 2 ν 2 γ 4 ν 4 γ 6 ν 6,.5 we obtan the followng addtonal condtons for the modfed RK method 2.2 to be of up to order fve see 16 : order requres: b γ 2 ;.6 order 4 requres n addton: b c γ 2, b a j γ 2 j ; j.7 order 5 requres n addton: b γ 2 b c a j γ 2 j, j ) 2, b γ 4, b a j c j γ 2 j, j b c 2 γ 2, b a j a jk γ 2. k j.8

6 6 Mathematcal Problems n Engneerng By smple calculaton, t s verfed that the new method s of algebrac order ffth. We denote the new method as MODRK5PLDPLAM. 4. Analyss of Stablty and Phase Propertes In ths secton, we are nterested n the stablty and phase propertes of the new method. Lambert and Watson s stablty theory 2 was reformulated by Coleman and Ixaru for the perodcty of exponentally ftted symmetrc methods for y f x, y. Van de Vyver 4 adapted ths theory to RK methods. Followng Van de Vyver s approach, we consder the test equaton as follows: y λy, λ >. 4.1 Applyng the modfed RK method 2.2 to test 4.1 yelds the dfference equaton y n 1 M θ, ν y n, θ λh, 4.2 where M θ, ν det I θa θγ ν b T) det I θa 4. wth I the s s dentty matrx. Defnton 4.1 see 4. For the modfed RK method 2.2 wth stablty functon M θ, ν, theregonntheθ-ν plane Ω : { θ, ν : M θ, ν 1} 4.4 s called the regon of magnary stablty. And any closed curve defned by M θ, ν 1sa stablty boundary of the method. In Fgure 1 we plot the regon of magnary stablty for the method MODRK5PLDPLAM. Defnton 4.2 see 4. For the modfed RK method 2.2 wth stablty functon M θ, ν, the quanttes P θ, ν θ arg M θ, ν, D θ, ν 1 M θ, ν 4.5

7 Mathematcal Problems n Engneerng 7 Stablty regon of method MODRK5PLDPLAM v Fgure 1: Regons of magnary stablty of the MODDPHARK5 method. θ are called the phase lag dsperson and amplfcaton factor error dsspaton, respectvely. If P θ, ν c φ θ q 1 O θ q ), D θ, ν c d θ p 1 O θ p ), 4.6 the method s sad to be of phase-lag order q and dsspaton order p, respectvely, where the c φ and c d are called the phase-lag constant and dsspaton constant, respectvely. We note that, by defnton, when ν θ ω λ, t must be true that P θ, ν and D θ, ν. In general, ω / λ snce the fttng frequency ω s just an estmate of the true frequency. Therefore the order of P θ, ν and D θ, ν ndefnton 4.2 measure to what extent a modfed RK method s accurate n phase and dsspaton. Denotng the rato r ν/θ ω/λ, we obtan the followng expressons for the phase lag and the dsspaton error of the new method MODRK5PLDPLAM: r P θ, 2 1 ) 2 ) r 2 rθ r D θ, 2 1 ) r 2) rθ 5244 θ 7 O θ 9), θ 6 O θ 8). 4.7 Thus, the method MODRK5PLDPLAM has a phase lag of order sx and a dsspaton of order fve. 5. Numercal Experments In ths secton, we test the numercal performance of the new ffth-order method n the ntegraton of the radal Schrödnger equaton wth the well-known Woods-Saxon potental,

8 8 Mathematcal Problems n Engneerng 8 Woods-Saxon potental wth E = log 1 ERR) N MODRK5PLDPLAM PHARK5S MODPHARK5S ARK5 MODPHARK5V PHADISRK5S Fgure 2: Effcency curves for E respectvely. We compare the new method wth some exstng hghly effcent methods n the lterature. The methods we choose for comparson are as follows: PHARK5S: the phase-ftted ffth-order RK method gven by Smos n 17, MODPHARK5S: the modfed phase-ftted ffth-order RK method gven by Smos and Aguar n 18, MODPHARK5V: the hgher-order method of the modfed phase-ftted embedded RK5 2.4 par gven by Van de Vyver n 16, v ARK5: an adapted ffth-order RK method gven by Fang et al. n 5, v PHADISRK5S: the phase-ftted and zero dsspaton ffth-order RK method gven by Tstouras and Smos n 22, v MODRK5PLDPLAM: the phase-ftted ffth-order method derved n ths paper. We consder the numercal ntegraton of the Schrödnger equaton 1.1 wth the well-known Woods-Saxon potental v x c z 1 a 1 z, 5.1 where z exp a x b 1 1,c 5, a 5/, b 7. The problem s solved n the nterval, 15. Followng 16, 6 8, we choose the fttng frequency ω { 5 E, x, 6.5, E, x 6.5,

9 Mathematcal Problems n Engneerng 9 7 Woods-Saxon potental wth E = log 1 ERR) N MODRK5PLDPLAM PHARK5S MODPHARK5S ARK5 MODPHARK5V PHADISRK5S Fgure : Effcency curves for E Woods-Saxon potental wth E = log 1 ERR) N MODRK5PLDPLAM PHARK5S MODPHARK5S ARK5 MODPHARK5V PHADISRK5S Fgure 4: Effcency curves for E In the numercal experment we consder the resonance problem E >, the numercal results E calculated are compared wth the analytcal soluton E analytcal of the Woods-Saxon potental, rounded to sx decmal places. In Fgures 2,, 4, and 5, we plot the error log 1 E analytcal E calculated versus N wth the ntegraton step-sze 1/2 N for E analytcal , , , and , respectvely.

10 1 Mathematcal Problems n Engneerng 4.5 Woods-Saxon potental wth E = log 1 ERR) N MODRK5PLDPLAM PHARK5S MODPHARK5S ARK5 MODPHARK5V PHADISRK5S Fgure 5: Effcency curves for E Conclusons and Dscussons Based on the classcal ffth RK method of Dormand and Prnce, a new optmzed explct modfed RK method wth modfyng parameters s obtaned by nullfyng the dsperson, the dsspaton, and the frst dervatve of the dsperson. The numercal results stated n Fgures 2 5 llustrate the hgher effcency of the new method compared to some hghly effcent methods n the recent lterature 16 18, 22, 5. Acknowledgments The authors are deeply grateful to the anonymous referees for ther constructve comments and valuable suggestons. Ths research s partally supported by NSFC no , the foundaton of Shandong Outstandng Young Scentsts Award Project no. BS21SF1,the foundaton of Scentfc Research Project of Shandong Unverstes no. J11LG69,andNSFof Shandong Provnce, Chna no. ZR211AL6. References 1 A. C. Allson, The numercal soluton of coupled dfferental equatons arsng from the Schrödnger equaton, Computatonal Physcs, vol. 6, pp , J. M. Blatt, Practcal ponts concernng the soluton of the Schrödnger equaton, Computatonal Physcs, vol. 1, pp , J. W. Cooley, An mproved egenvalue corrector formula for solvng the Schrödnger equaton for central felds, Mathematcs of Computaton, vol. 15, pp. 6 74, G. Avdelas, T. E. Smos, and J. Vgo-Aguar, An embedded exponentally-ftted Runge-Kutta method for the numercal soluton of the Schrödnger equaton and related per-odc ntal-value problems, Computer Physcs Communcatons, vol. 11, pp , 2.

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