Appl. Math. Inf. Sci. 8, No. 2, 447 -458 (2014) 447 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080201 On the Explicit Four-Step Methods with Vanished Phase-Lag and its First Derivative T. E. Simos 1,2,∗ 1 Department of Mathematics, College of Sciences, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia 2 Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, GR-221 00 Tripolis, Greece Received: 5 Mar. 2013, Revised: 6 Jul. 2013, Accepted: 8 Jul. 2013 Published online: 1 Mar. 2014 Abstract: In the present paper, we will investigate a family of explicit four-step methods first introduced by Anastassi and Simos [1] for the case of vanishing of phase-lag and its first derivative. These methods are efficient for the numerical solution of the Schr¨ odinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions. As we mentioned before the main scope of this paper is the study of the elimination of the phase-lag and its first derivative of the family of the methods to be investigated. A comparative error and stability analysis will be presented for the studied family of four-step explicit methods. The new obtained methods will finally be tested on the resonance problem of the Schr ¨ odinger equation in order to examine their efficiency. Keywords: Phase-lag, initial value problems, oscillating solution, symmetric, multistep, Schr¨ odinger equation 1 Introduction In this paper the numerical solution of special second-order initial-value problems of the form y ′′ (x)= f (x, y), y(x 0 )= y 0 and y ′ (x 0 )= y ′ 0 (1) with an periodical and/or oscillatory solutions are investigated. The main characteristic of the mathematical models of the above mentioned problems is that the ordinary differential equations which describe the above models are of second order in which the first derivative y ′ does not appear explicitly (see for numerical methods for these problems [3]-[32] and references therein). 2 Analysis of the Phase-lag For Symmetric Multistep Methods For the numerical integration of the above mentioned initial value problem (1), one can use multistep methods of the form k ∑ i=0 c i y n+i = h 2 k ∑ i=0 b i f (x n+i , y n+i ) (2) with k steps over the equally spaced intervals {x i } k i=0 ∈ [a, b] and h = |x i+1 − x i |, i = 0(1)k − 1, where h is called stepsize of integration. If the method is symmetric then c i = c k−i and b i = b k−i , i = 0(1)⌊ k 2 ⌋. The Multistep Method (2) is associated with the operator L(x)= k ∑ i=0 c i u(x + ih) − h 2 k ∑ i=0 b i u ′′ (x + ih) (3) where u ∈ C 2 . Definition 1.[1] The multistep method (2) is called algebraic of order q if the associated linear operator L vanishes for any linear combination of the linearly independent functions 1, x, x 2 ,..., x q+1 . Application of a symmetric 2 m-step method, that is for i = −m(1)m, to the scalar test equation y ′′ = −φ 2 y (4) leads to the following difference equation: A m (v) y n+m + ... + A 1 (v) y n+1 + A 0 (v) y n +A 1 (v) y n−1 + ... + A m (v) y n−m = 0 (5) ∗ Corresponding author e-mail: [email protected]c 2014 NSP Natural Sciences Publishing Cor.
Applied Mathematics & Information SciencesAn International Journal
http://dx.doi.org/10.12785/amis/080201
On the Explicit Four-Step Methods with VanishedPhase-Lag and its First Derivative
T. E. Simos1,2,∗
1 Department of Mathematics, College of Sciences, King Saud University,P. O. Box 2455, Riyadh 11451, Saudi Arabia2 Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and
Informatics, University of Peloponnese, GR-221 00 Tripolis, Greece
Received: 5 Mar. 2013, Revised: 6 Jul. 2013, Accepted: 8 Jul. 2013Published online: 1 Mar. 2014
Abstract: In the present paper, we will investigate a family of explicit four-step methods first introduced by Anastassi and Simos [1]for the case of vanishing of phase-lag and its first derivative. Thesemethods are efficient for the numerical solution of the Schrodingerequation and related initial-value or boundary-value problems with periodicand/or oscillating solutions. As we mentioned before themain scope of this paper is the study of the elimination of the phase-lag and its first derivative of the family of the methods to beinvestigated. A comparative error and stability analysis will be presented for the studied family of four-step explicit methods. The newobtained methods will finally be tested on the resonance problem of the Schrodinger equation in order to examine their efficiency.
In this paper the numerical solution of specialsecond-order initial-value problems of the form
y′′(x) = f (x,y), y(x0) = y0 and y′(x0) = y′0 (1)
with an periodical and/or oscillatory solutions areinvestigated. The main characteristic of the mathematicalmodels of the above mentioned problems is that theordinary differential equations which describe the abovemodels are of second order in which the first derivativey′
does not appear explicitly (see for numerical methods forthese problems [3] - [32] and references therein).
2 Analysis of the Phase-lag For SymmetricMultistep Methods
For the numerical integration of the above mentionedinitial value problem (1), one can use multistep methodsof the form
k
∑i=0
ciyn+i = h2k
∑i=0
bi f (xn+i ,yn+i) (2)
with k steps over the equally spaced intervals{xi}ki=0 ∈
[a,b] andh= |xi+1−xi |, i = 0(1)k−1, whereh is calledstepsize of integration.If the method is symmetric thenci = ck−i andbi = bk−i ,i = 0(1)⌊ k
2⌋.
The Multistep Method (2) is associated with the operator
L(x) =k
∑i=0
ciu(x+ ih)−h2k
∑i=0
biu′′(x+ ih) (3)
whereu∈C2.
Definition 1.[1] The multistep method (2) is calledalgebraic of orderq if the associated linear operatorLvanishes for any linear combination of the linearlyindependent functions 1, x, x2, . . . , xq+1.
Application of a symmetric 2m-step method, that is fori =−m(1)m, to the scalar test equation
448 T. E. Simos: On the Explicit Four-Step Methods with Vanished...
wherev= φ h, h is the step length andA j(v) j = 0(1)marepolynomials ofv.
The associated with (5) characteristic equation is givenby:
Am(v)λ m+ ...+A1(v)λ +A0(v)
+A1(v)λ−1+ ...+Am(v)λ−m = 0 (6)
Lambert and Watson (1976) introduce the followingdefinitions:
Definition 2.A symmetric 2m-step method withcharacteristic equation given by (6) is said to have aninterval of periodicity (0,v2
0) if, for all v ∈ (0,v20), the
rootsλi , i = 1(1)2m of Eq. (6) satisfy:
λ1 = eiθ(v), λ2 = e−iθ(v), and |λi | ≤ 1, i = 3(1)2m (7)
whereθ(v) is a real function of v.
Definition 3.[14], [ 15] For any method corresponding tothe characteristic equation (6) the phase-lag is defined asthe leading term in the expansion of
t = v−θ(v) (8)
Then if the quantity t= O(vp+1) as v→ ∞, the order ofphase-lag is p.
Definition 4.[2] A method for which the phase-lagvanishes, is calledphase-fitted
Theorem 1.[14] The symmetric2m-step method withcharacteristic equation given by (6) has phase-lag orderp and phase-lag constant c given by
The formula proposed from the above theorem gives us adirect method to calculate the phase-lag of any symmetric2m- step method.
In our case, the symmetric four-step method has phase-lagorderp and phase-lag constantc given by:
−cvp+2+O(vp+4) = T0T1
(10)
T0 = 2A2(v) cos(2v)+2A1(v) cos(v)+A0(v)
T1 = 8A2(v)+2A1(v)
3 The Family of Explicit Four-Step Methodswith Vanished Phase-Lag and Its FirstDerivative
From the form (2) and without loss of generality weassumeck = 1 and we can write
yn+k+k−1
∑i=0
ci yn+i = h2k
∑i=0
bi f (xn+i ,yn+i), (11)
If the method is symmetric thenci = ck−i andbi = bk−i , i = 0(1)⌊ k
2⌋.
From the form (11) with k = 4 andb4 = 0 we get theform of the explicit symmetric four-step methods [1]:
yn+4+c3 (yn+3+yn+1)+c2yn+2+yn =
= h2
[
b1 ( fn+3+ fn+1)+b0 fn+2
]
(12)
where fi = y′′ (xi ,yi) , i = n−1(1)n+1.
3.1 First Case
Considering (12), we choose:
c0 = −2c1−2, c1 = − 110
(13)
The choice was based on the results of the investigationpresented in [1]. Based on this study, the above values givefor the method (12) the higher accuracy.
Demanding now the above method to have the phase-lag and its first derivative vanished, the following systemof equations is produced:
T2395 +2v2b1
= 0
− T3
(10v2b1+39)2 = 0 (14)
where
T2 = 2 cos(
2v)
+2(
− 110
+v2b1
)
cos(
v)
− 95+v2b0
T3 = 100 sin(
v)
v4b12+400 sin
(
v)
cos(
v)
v2b1
+ 380 sin(
v)
v2b1+400vb1
(
cos(
v))2
− 800vb1 cos(
v)
+1560 sin(
v)
cos(
v)
− 390vb0−380vb1−39 sin(
v)
Solving the above system of equations, we obtain thecoefficients of the new proposed method:
The following Taylor series expansions should be usedin the cases that the formulae given by (15) are subject toheavy cancellations for some values of|v| :
b0 =54+
161600
v2− 154120160
v4
+473
67200v6− 28811
79833600v8
+3991
342144000v10− 1023167
3487131648000v12
+5317391
1778437140480000v14− 60114757
243290200817664000v16
− 21637421912164510040883200000
v18+ . . .
b1 =5340
− 1611200
v2+943
201600v4
− 13134400
v6+109
798336000v8
− 4574790016000
v10− 280333170119680000
v12
− 32029313556874280960000
v14− 31704163347557429739520000
v16
− 1574039107170303140572364800000
v18+ . . . (16)
The behavior of the coefficients is given in thefollowing Figure 1.
The new obtained method (12) (mentioned asFourStepI) with the coefficients given by (15)-(16) has alocal truncation error which is given by:
LTEFourStepI=161h6
2400
(
y(6)n +2φ2y(4)n +φ4y(2)n
)
+O(
h8)
(17)
3.2 Second Case
Considering (12), we choose:
c0 = −2c1−2, c1 = − 920
(18)
The choice was based on the results of the investigationpresented in [1]. Based on this study, the above values givefor the method (12) the medium accurate solution.
Requesting the above method to have the phase-lagand its first derivative equal to zero, the following systemof equations is produced:
T67110+2v2b1
= 0
− T7
(20v2b1+71)2 = 0 (19)
Fig. 1: Behavior of the coefficients of the new proposed methodgiven by (15) for several values ofv= φ h.
450 T. E. Simos: On the Explicit Four-Step Methods with Vanished...
The coefficients of the new proposed method areobtained as solution of the above system of equations :
b0 =110
T8
sin(v)v3 , b1 = 1/20T9
sin(v)v3 (20)
where
T8 = 41 sin(v)v+10v sin(3v)−9 cos(2v)
− 2 cos(v)+20 cos(3v)−9
T9 = 9 sin(v)v−40v sin(2v)
− 40 cos(2v)+18 cos(v)+22
The following Taylor series expansions should be usedin the cases that the formulae given by (20) are subject toheavy cancellations for some values of|v| :
b0 =2324
+3291200
v2− 103913440
v4
+25339
3628800v6− 11719
31933440v8
+331
29952000v10− 7440421
20922789888000v12
− 117812413556874280960000
v14− 1104954712476420554752000
v16
− 602036183372987060245299200000
v18+ . . .
b1 =311240
− 3292400
v2+589
134400v4
− 9197257600
v6− 46191596672000
v8
− 1670341513472000
v10− 8357161209227898880000
v12
− 288218997113748561920000
v14− 452918911033569198080000
v16
− 424990502891021818843434188800000
v18+ . . . (21)
The behavior of the coefficients is given in thefollowing Figure 1.
The new obtained method (12) (mentioned asFourStepII) with the coefficients given by (20)-(21) has alocal truncation error which is given by:
LTEFourStepII=329h6
4800
(
y(6)n +2φ2y(4)n +φ4y(2)n
)
+O(
h8)
(22)
4 Comparative Error Analysis
We will study the following methods (Case I)1:
1 The results for the Case II is analogous
Fig. 2: Behavior of the coefficients of the new proposed methodgiven by (20) for several values ofv= φ h.
4.1 Classical Method(i.e. the method (12) withconstant coefficients of the Case I)
LTECL =161h6
2400y(6)n +O
(
h8) (23)
4.2 The Method with Vanished Phase-LagProduced in [1]
4.3 The New Proposed Method with VanishedPhase-Lag and its First Derivative Produced inSection 3
LTEFourStepII=161h6
2400
(
y(6)n +2φ2y(4)n +φ4y(2)n
)
+O(
h8)
(25)The procedure contains the following stages
–Computation of the derivatives which presented in theformulae of the Local Truncation Errors. Thesecomputations lead to the following formulae:
y(2)n = (V(x)−Vc+G) y(x)
y(3)n =
(
ddx
g(x)
)
y(x)+(g(x)+G)ddx
y(x)
y(4)n =
(
d2
dx2 g(x)
)
y(x)+2
(
ddx
g(x)
)
ddx
y(x)
+(g(x)+G)2y(x)
y(5)n =
(
d3
dx3 g(x)
)
y(x)+3
(
d2
dx2 g(x)
)
ddx
y(x)
+4 (g(x)+G)y(x)ddx
g(x)
+(g(x)+G)2 ddx
y(x)
y(6)n =
(
d4
dx4 g(x)
)
y(x)+4
(
d3
dx3 g(x)
)
ddx
y(x)
+7 (g(x)+G)y(x)d2
dx2 g(x)
+4
(
ddx
g(x)
)2
y(x)
+6 (g(x)+G)
(
ddx
y(x)
)
ddx
g(x)
+(g(x)+G)3y(x)
y(7)n =
(
d5
dx5 g(x)
)
y(x)+5
(
d4
dx4 g(x)
)
ddx
y(x)
+11(g(x)+G)y(x)d3
dx3 g(x)
+15
(
ddx
g(x)
)
y(x)d2
dx2 g(x)
+13(g(x)+G)
(
ddx
y(x)
)
d2
dx2 g(x)
+10
(
ddx
g(x)
)2 ddx
y(x)
+9 (g(x)+G)2y(x)ddx
g(x)+(g(x)+G)3 ddx
y(x)
y(8)n =
(
d6
dx6 g(x)
)
y(x)+6
(
d5
dx5 g(x)
)
ddx
y(x)
+16(g(x)+G)y(x)d4
dx4 g(x)+26
(
ddx
g(x)
)
y(x)
d3
dx3 g(x)+24(g(x)+G)
(
ddx
y(x)
)
d3
dx3 g(x)
+15
(
d2
dx2 g(x)
)2
y(x)+48
(
ddx
g(x)
)
(
ddx
y(x)
)
d2
dx2 g(x)+22(g(x)+G)2y(x)
d2
dx2 g(x)+28(g(x)+G)y(x)
(
ddx
g(x)
)2
+12(g(x)+G)2(
ddx
y(x)
)
ddx
g(x)
+(g(x)+G)4y(x)
. . .
–Based on the above formulae and substituting them inthe expressions of the Local Truncation Error we canproduce formulae of the local errors which aredependent from the energyE.
–We study two cases in terms of the value ofE withinthe Local Truncation Error analysis :
1.The Energy is close to the potential, i.e.,G = Vc −E ≈ 0. Consequently, the free terms ofthe polynomials inG are considered only. Thus,for these values ofG, the methods are ofcomparable accuracy. This is because the freeterms of the polynomials inG are the same for thecases of the classical method and of the methodswith vanished the phase-lag and its derivatives.
2.G>> 0 orG<< 0. Then|G| is a large number.
–Finally we compute the asymptotic expansions of theLocal Truncation Errors
The following asymptotic expansions of the LocalTruncation Errors are obtained based on the analysispresented above :
452 T. E. Simos: On the Explicit Four-Step Methods with Vanished...
4.5 The Method with Vanished Phase-LagProduced in [1]
LTEMethAnasSim= h6
(
1612400
g(x)y(x) G2+ · · ·)
+O(
h8)
(27)
4.6 The New Proposed Method with VanishedPhase-Lag and its First Derivative Produced inSection 3
LTEFourStepII= h6
[(
1612400
(g(x))2y(x)
+1611200
(
ddx
g(x)
)
ddx
y(x)+161480
(
d2
dx2 g(x)
)
y(x)
)
G+ · · ·]
+O(
h8)(28)
From the above equations we have the followingtheorem:
Theorem 2.For the Classical Four-Step Explicit Method,the error increases as the third power of G. For theFour-Step Explicit Phase-Fitted Method developed in [1], the error increases as the second power of G. Finally,for the new obtained Four-Step Explicit Method withVanished Phase-lag and its First Derivative, the errorincreases as the first power of G. So, for the numericalsolution of the time independent radial Schrodingerequation the New Proposed Method with VanishedPhase-Lag and its First Derivative is the most efficientfrom theoretical point of view, especially for large valuesof |G|= |Vc−E|.
5 Stability Analysis
In order to investigate the stability of the new developedmethods, we apply them to the scalar test equation:
y′′ =−ω2y. (29)
This leads to the following difference equation:
A2 (s,v) (yn+2+yn−2) + A1 (s,v) (yn+1+yn−1)
+ A0 (s,v) yn = 0 (30)
where
A2 (s,v) = 1, A0 (s,v) = − 110
+110
T10
sin(v)v3
A0 (s,v) = −95+
15
T11
sin(v)v3 (31)
where
T10 = s2(
−20v sin(
2v)
+18−20 cos(
2v)
+sin(
v)
v+2 cos(
v))
T11 = s2(
24 sin(
v)
v+5v sin(
3v)
−cos(
2v)
−8 cos(
v)
+10 cos(
3v)
−1)
ands= ω h.
Remark.The frequency of the scalar test equation (29), ω,is not equal with the frequency of the scalar test equation(4), φ , i.e.ω 6= φ .
Based on the analysis presented in Section 2, we havethe following definitions:
Definition 5.(see [16]) A method is called P-stable if itsinterval of periodicity is equal to(0,∞).
Definition 6.A method is called singularly almostP-stable if its interval of periodicity is equal to(0,∞)−S2
only when the frequency of the phase fitting is the same asthe frequency of the scalar test equation, i.e. s= v.
In Figure 3 we present thes− v plane for the methoddeveloped in this paper (First Case). In Figure 4 wepresent thes− v plane for the method developed in thispaper (Second Case).
Remark.A shadowed area denotes thes− v region wherethe method is stable, while a white area denotes the regionwhere the method is unstable.
Remark.There are cases where it is appropriate to observethe surroundings of the first diagonal of the s− vplane. These cases have mathematical models where inorder to apply the new produced methods the frequencyof the phase fitting must be equal to the frequency of thescalar test equation. The cases are many problems insciences and engineering (for example the timeindependent Schrodinger equation).
Based on the above remark, the case where thefrequency of the scalar test equation is equal with thefrequency of phase fitting is now studied, i.e. weinvestigate the case wheres= v (i.e. see the surroundingsof the first diagonal of thes− v plane). Based on thisstudy we extract the results that the interval of periodicity
Fig. 3: s−v plane of the the new obtained method with vanishedphase-lag and its first derivative (First Case)
Fig. 4: s−v plane of the the new obtained method with vanishedphase-lag and its first derivative (Second Cas)
of the new methods developed in section 3 are equal to:(0,8.0) (First Case) and(0,9.9) (Second Case).
The above investigation leads to the followingtheorem:
Theorem 3.The methods developed in section 3:
–are of fourth algebraic order,–have the phase-lag and its first derivative equal to zero–have an interval of periodicity equals to:(0,8.0) (FirstCase) and(0,9.9) (Second Case) when the frequencyof the scalar test equation is equal with the frequencyof phase fitting
6 Numerical results
The efficiency of the new proposed explicit four-stepmethods is investigated via the approximate solution ofthe radial time-independent Schrodinger equation.
The one-dimensional time independent Schrodingerequation with mathematics model given by :
y′′(r) = [l(l +1)/r2+V(r)−k2]y(r). (32)
is a boundary value problem which has the followingboundary conditions :
y(0) = 0 (33)
and another boundary condition, for large values ofr,determined by physical properties of the specific problem.
We give the following definitions of the functions,quantities and parameters for the above mathematicalmodel (32) :
1.The functionW(r) = l(l + 1)/r2 +V(r) is calledtheeffective potential. This satisfiesW(x)→ 0 asx→ ∞,
2.The quantityk2 is a real number denotingthe energy,3.The quantity l is a given integer representing the
angular momentum,4.V is a given function which denotes thepotential.
In order the new obtained methods to be applied toany problem, and since these methods are frequencydependent methods, the value of parameterφ (see forexample the notation after (4) and the formulae in section3) must be defined. The parameterφ for the case of theradial Schrodinger equation is given by (forl = 0) :
φ =√
|V (r)−k2|=√
|V (r)−E| (34)
whereV (r) is the potential andE is the energy.
6.1 Woods-Saxon potential
For the purpose of our numerical tests we use the wellknown Woods-Saxon potential. This can be written as :
454 T. E. Simos: On the Explicit Four-Step Methods with Vanished...
with q= exp[
r−X0a
]
, u0 =−50, a= 0.6, andX0 = 7.0.
The behavior of Woods-Saxon potential is shown inFigure 5.
-50
-40
-30
-20
-10
02 4 6 8 10 12 14
r
The Woods-Saxon Potential
Fig. 5: The Woods-Saxon potential.
For some potentials, such as the Woods-Saxonpotential, and using their investigation, some criticalpoints are defined and these points are used in order theparameterφ to be defined (see for details [36]).
For the purpose of our tests, it is appropriate to chooseφ as follows (see for details [37] and [38]):
φ =
√−50+E, for r ∈ [0,6.5−2h],√−37.5+E, for r = 6.5−h√−25+E, for r = 6.5√−12.5+E, for r = 6.5+h√E, for r ∈ [6.5+2h,15]
(36)
For example, in the point of the integration regionr =6.5−h, the value ofφ is equal to:
√−37.5+E. So,w =
φ h=√−37.5+E h. In the point of the integration region
r = 6.5−3h, the value ofφ is equal to:√−50+E, etc.
6.2 Radial Schrodinger Equation - TheResonance Problem
We will study the numerical solution of the radial timeindependent Schrodinger equation (32) with theWoods-Saxon potential (35) for the examination of theefficiency of the new proposed methods. Theapproximation of the true (infinite) interval of integrationby a finite one is necessary for the approximate solutionof this problem. We take the integration intervalr ∈ [0,15] for the purposes of our numerical experiments.
Fig. 6: Accuracy (Digits) for several values ofCPU Time (inSeconds) for the eigenvalueE2 = 341.495874. The nonexistenceof a value of Accuracy (Digits) indicates that for this value ofCPU, Accuracy (Digits) is less than 0
We consider equation (32) in a rather large domain ofenergies, i.e.,E ∈ [1,1000].
In the case of positive energies,E = k2, the potentialdecays faster than the terml(l+1)
r2 and the Schrodingerequation effectively reduces to
y′′ (r)+
(
k2− l(l +1)r2
)
y(r) = 0 (37)
for r greater than some valueR.The above equation has linearly independent solutions
kr j l (kr) and krnl (kr), where j l (kr) and nl (kr) are thespherical Bessel and Neumann functions respectively.Thus, the solution of equation (32) (whenr → ∞), has theasymptotic form
whereδl is the phase shift that may be calculated from theformula
tanδl =y(r2)S(r1)−y(r1)S(r2)
y(r1)C(r1)−y(r2)C(r2)(39)
for r1 andr2 distinct points in the asymptotic region (wechooser1 as the right hand end point of the interval ofintegration andr2 = r1 − h) with S(r) = kr j l (kr) andC(r) = −krnl (kr). Since the problem is treated as aninitial-value problem, we needy j , j = 0,(1)3 beforestarting a four-step method. From the initial condition, weobtainy0. The valuesyi , i = 1(1)3 are obtained by usinghigh order Runge-Kutta-Nystrom methods(see [39] and[40]). With these starting values, we evaluate atr2 of theasymptotic region the phase shiftδl .
For positive energies, we have the so-called resonanceproblem. This problem consists either of finding thephase-shiftδl or finding thoseE, for E ∈ [1,1000], atwhich δl = π
2 . We actually solve the latter problem,known asthe resonance problem.
The boundary conditions for this problem are:
y(0) = 0, y(r) = cos(√
Er)
for larger. (40)
We compute the approximate positive eigenenergies ofthe Woods-Saxon resonance problem using:
–The eighth order multi-step method developed byQuinlan and Tremaine [41], which is indicated asMethod QT8.
–The tenth order multi-step method developed byQuinlan and Tremaine [41], which is indicated asMethod QT10.
–The twelfth order multi-step method developed byQuinlan and Tremaine [41], which is indicated asMethod QT12.
–The fourth algebraic order method of Chawla and Raowith minimal phase-lag [43], which is indicated asMethod MCR4
–The exponentially-fitted method of Raptis and Allison[42], which is indicated asMethod RA
–The hybrid sixth algebraic order method developed byChawla and Rao with minimal phase-lag [44], whichis indicated asMethod MCR6
–The classical form of the fourth algebraic order four-step method developed in Section 3, which is indicatedasMethod NMCL 3.
–The Phase-Fitted Method (Case 1) developed in [1],which is indicated asMethod NMPF1
–The Phase-Fitted Method (Case 2) developed in [1],which is indicated asMethod NMPF2
–The New Obtained Method developed in Section 3(Case 2), which is indicated asMethod NMC2
–The New Obtained Method developed in Section 3(Case 1), which is indicated asMethod NMC1
3 with the term classical we mean the method of Section 3 withconstant coefficients
Fig. 7: Accuracy (Digits) for several values ofCPU Time (inSeconds) for the eigenvalueE2 = 341.495874. The nonexistenceof a value of Accuracy (Digits) indicates that for this value ofCPU, Accuracy (Digits) is less than 0
The numerically calculated eigenenergies arecompared with reference values4. In Figures 8 and 9, wepresent the maximum absolute errorErrmax= |log10(Err) | where
Err = |Ecalculated−Eaccurate| (41)
of the eigenenergies E2 = 341.495874 andE3 = 989.701916 respectively, for several values of CPUtime (in seconds). We note that the CPU time (in seconds)counts the computational cost for each method.
4 the reference values are computed using the well known two-step method of Chawla and Rao [44] with small step size for theintegration
456 T. E. Simos: On the Explicit Four-Step Methods with Vanished...
7 Conclusions
In this paper, we studied a family of explicit four-stepmethods first introduced by Anastassi and Simos [1]. Themain purpose of this investigation is the study of theelimination of the phase-lag and its first derivative of theabove mentioned family of the methods. For this familywe presented a comparative error and stability analysis.We have also investigated the effect of the vanishing ofthe phase-lag and its first derivative on the efficiency ofthe above mentioned methods for the approximatesolution of the radial Schrodinger equation and relatedproblems.
From the results presented above, we can make thefollowing remarks:
1.The classical form of the tenth algebraic orderfour-step multiderivative method developed in Section3, which is indicated asMethod NMCL is moreefficient than the fourth algebraic order method ofChawla and Rao with minimal phase-lag [43], whichis indicated asMethod MCR4. Both the abovementioned methods are more efficient than theexponentially-fitted method of Raptis and Allison[42], which is indicated asMethod RA.
2.The tenth algebraic order multistep method developedby Quinlan and Tremaine [41], which is indicated asMethod QT10 is more efficient than the fourthalgebraic order method of Chawla and Rao withminimal phase-lag [43], which is indicated asMethodMCR4. TheMethod QT10 is also more efficient thanthe eighth order multi-step method developed byQuinlan and Tremaine [41], which is indicated asMethod QT8. Finally, the Method QT10 is moreefficient than the hybrid sixth algebraic order methoddeveloped by Chawla and Rao with minimalphase-lag [44], which is indicated asMethod MCR6for large CPU time and less efficient than theMethodMCR6 for small CPU time.
3.The twelfth algebraic order multistep methoddeveloped by Quinlan and Tremaine [41], which isindicated asMethod QT12 is more efficient than thetenth order multistep method developed by Quinlanand Tremaine [41], which is indicated asMethodQT10
4.The Phase-Fitted Method (Case 1) developed in [1],which is indicated asMethod NMPF1 is moreefficient than the classical form of the fourth algebraicorder four-step method developed in Section 3, whichis indicated as Method NMCL , theexponentially-fitted method of Raptis and Allison [42]and the Phase-Fitted Method (Case 2) developed in[1], which is indicated asMethod NMPF2
5.The New Obtained Method developed in Section 3(Case 2), which is indicated asMethod NMC2 ismore efficient than the classical form of the fourthalgebraic order four-step method developed in Section3, which is indicated asMethod NMCL , the
exponentially-fitted method of Raptis and Allison [42]and the Phase-Fitted Method (Case 2) developed in[1], which is indicated asMethod NMPF2 and thePhase-Fitted Method (Case 1) developed in [1], whichis indicated asMethod NMPF1
6.The New Obtained Method developed in Section 3(Case 2), which is indicated asMethod NMC1 is themost efficient one.
All computations were carried out on a IBM PC-ATcompatible 80486 using double precision arithmetic with16 significant digits accuracy (IEEE standard).
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458 T. E. Simos: On the Explicit Four-Step Methods with Vanished...
Theodore E. Simos(b. 1962 in Athens, Greece).He holds a Ph.D. onNumerical Analysis (1990)from the Departmentof Mathematicsof the National TechnicalUniversity of Athens,Greece. He is Highly CitedResearcher in Mathematics
(http://isihighlycited.com/), Active Member of theEuropean Academy of Sciences and Arts, Active Memberof the European Academy of Sciences and CorrespondingMember of European Academy of Sciences, Arts andLetters. He is Senior Editor of the Journal: AppliedMathematics and Computation (Elsevier, INC),Editor-in-Chief of three scientific journals and editor ofmore than 25 scientific journals. He is reviewer in severalother scientific journals and conferences. His researchinterests are in numerical analysis and specifically innumerical solution of differential equations, scientificcomputing and optimization. He is the author of over 400peer-reviewed publications and he has more than 2000citations (excluding self-citations).
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