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Error analysis of exponential-fitted methods for the numerical solution of the one-dimensional Schrödinger equation
Physics Letters A 177 ( 1993) 345-350 North-Holland
PHYSICS
LETTERS
A
Error analysis of exponential-fitted methods for the numerical solution of the one-dimensional Schredinger equation T.E. Simos Informatics Laboratory, Agricultural University ofAthens, Iera Odos 75. Athens 11855, Greece
Received 14 July 1992; accepted for publication 2 1 April 1993 Communicated by D.D. Holm
In this paper an error analysis of two-step and four-step methods for the numerical integration of the one-dimensional Schrijdinger equation is developed. The conclusion from this analysis is that the two-step Runge-Kutta type exponential fitted method by Cash, Raptis and Simos and the four-step exponential fitted method by Simos are among the now known exponential-fitted methods for the numerical solution of the radial Schriidinger equation the most effective ones.
1. Introduction There has always been a great activity from the computational point of view concerning the radial Schrodinger equation, the aim being to achieve a fast and reliable algorithm that generates a numerical solution. The radial Schrijdinger equation has the form
Y” (x)
=f(x)Y(x)
2
(1.1)
where f(x) = U(x) -E and U(x) =1(1+ 1 )/x2+ V(x) is an effective potential for which U(x)+0 as x+co. The boundary conditions are y(0) =O and another condition when x-*cr, which depends on the physical considerations. The most popular method for the numerical solution of ( 1.1) has been Numerov’s one. Raptis and Allison [ 11 based on the Chebyshevian multistep theory of Lyche [2] have developed a Numerov-type exponential fitted method and applied it to the Schrodinger equation corresponding to positive energies (i.e. solution in the asymptotic region). This method integrates exactly the polynomials 1, x, x2, exp( +vx). Based on this approach Ixaru and Rizea [ 31 have constructed an exponential-fitted method which integrates exactly the polynomials 1, x, exp( k vx), xexp( k VX). Also Raptis [ 41 has developed an exponential-fitted method which integrates exactly the polynomials exp ( k vx) , x exp ( 31ux) , x2 exp ( + vx) . We note that all these methods are fourthorder algebraic. Raptis [ 5,6] based on the classical method of Henrici [ 71 developed later a four-step sixth-order exponential-fitted method which integrates exactly the polynomials 1, x, x2, x3, x4, x5, exp ( ? vx) and 1, x, exp ( 2 vx), x exp ( f vx), x2 exp ( + vx) respectively and then Simos [ 8 ] developed an exponential-fitted method which integrates exactly the polynomials exp( 2 VX), x exp ( k vx), x2 exp( + VX), x3 exp( k vx). Cash and Raptis [ 93 developed a sixth-order hybrid method for the numerical solution of ( 1.1) which integrates exactly the polynomials 1, x, x2, x3, x4, x5, x6, x7. Raptis and Cash [lo] have developed on the other hand an exponential-fitted sixth-order hybrid method which integrates exactly the polynomials 1, x, x2, x3, x4, x5, exp( + vx). Finally Cash, Raptis and Simos [ 111 developed a sixth-order exponential-fitted method which integrates exactly the polynomials 1, x, x2, x3, exp( + VX), xexp( + VX). The purpose of this paper is to examine theoretically these methods and find a quantitative estimation for 03759601/93/$
06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
345
Volume 177, number 4,5
PHYSICS LETTERS A
21 June 1993
the extent of the accuracy gain to be expected from the optimal versions. We must note that an error analysis for three of the four-step methods and for three of the four two-step methods has been done by Ixaru and Rizea 13,121.
2. Error analysis 2.1. Two-step methods and hybrid methods The local truncation following.
errors of the above-mentioned
two-step methods
and hybrid
methods
is given by the
Two-step methods ( 1) Numerov’s
method:
(2.1)
LTE,.,=-&h6y”(x)+O(h8), (2) Method
[ 1] :
of Raptis and Allison
(2.2)
LTERA=-~h6[yvi(x)+~2yiv(~)]+O(h8), ( 3 ) Method
of Ixaru and Rizea
[ 3] :
LTE,R=-&,h6[y’i(x)-2u2yi’(x)+v4yii(x)]+O(h8), (4) Method
of Raptis
(2.3)
[4]:
LTEn = - &h6[yvi(x)-3v2yiV(x)+3v4yii(x)-zfj]+O(h*).
(2.4)
Hybrid methods ( 1) Method
of Cash and Raptis
LTEcR = - &h8yviii(x) (2 ) Method
[ 91:
+O(h’O)
,
of Raptis and Cash [ 10 ] :
LTERc = - ~oh8[yviii(x)-v2yyi(x)]+O(h10), (3) Method
(2.6)
of Cash, Raptis and Simos [ 111:
L=CRS = - ~h8[yviii(x)-3v2yVi(~)+3u4yiV(~)-v6yii(~)]+O(h’0). 120960 These formulae are still not suffkiently transparent to enable drawing quantitative conclusions of each method. To do this we applied them to the one-dimensional Schrijdinger equation. Based on the work of Ixaru and Rizea [ 12 1, we write first f(x) of ( 1.1) in the form f(x) =g(x)
(2.5)
+d 3
(2.7) on the merits
(2.8)
where g(x) = U(x) - U, =g, where U, is the constant approximation of the potential previously mentioned, and d= v2= UC-E. Now g(x) depends on the potential and its constant approximation and d stands for the energy dependence. Then we express the derivatives y”(x), y”‘(x), .... ye”‘(x), which are terms of the local error estimation for346
Volume
177, number
PHY SKIS LETTERS
4,5
21 June 1993
A
mula, in terms of the equation i.e. y” =f(x)y(x). We note that g(“) (x) = U(“) (x) for any nth order derivative with respect to x. We also express the terms as polynomials of d. Finally, we introduce these expressions in the local error estimation formulae (2.1)-( 2.7 ) and we obtain the following expressions. Two-step methods
( 1) Numerov’s method: LTEN = h 6ERN , ,
ER ,=-~~{y[d3+3d2g+d(3g2+7U(2))+g3+4U’2+7gU(2)+U(4)]+y’(6dU’+6gU’+4U(3)))
(2.9)
(2) Method of Raptis and Allison [ 11:
LTERA= h6ER,
,
(2.10)
ERRA=-~{y[d2g+d(2g2+6U(2))+g3+4U’2+7gU(2)+U(4)]+y’(4dU’+6gU’+4U(3))},
(3) Method of Ixaru and Rizea [ 31: LTEIR = h 6ER,R , ERIR =-~{y[d(2g2+5U(2’)+g3+4U’2+7gU’2’+U(4’]+y’(2dU’+6gIT’+4U(3))}
,
(2.11)
(4) Method of Raptis [4]:
LTER = h 6ERR , (2.12)
ER R=-&{y[4dU(2)+g3+4U’2+7gU(2)+U(4)]+y’(6gU’+4U0))}. Hybrid methods
( 1) Method of Cash and Raptis [ 91: LTEcR = h ‘ERCR , ERcR = - &{d4y+4d3gy+d2[ +d[ (4U’3’+28U’
(6U2+22Uc2’)y+
+44Uc2’g+
+ [ (g4+28U’2g+22U(2)g2+
+ (24U’3’g+48U’2’U’
16Ut4’)y+24(
12U’y’]
U’g+ Uc3))y’]
16V4’g+ U’6’+26U’3’U’ + 15U’2’)y
+ 12U’g2+6U’5’)y’]},
(2.13)
(2) Method of Raptis and Cash [ 10 1: LTERc = h *ERRC, ERRc = - &{d3gy+d2[
(3g2+ 15U’2’)y+6U’y’]
+d[ (3g3+24U’2+37gU’2’+
15U’4’)y+
+ [ (g4+28U’2g+22U(2)g2+
16Uc4’g+ U’6’+26U’3’U’
+ (24U’3’g+48U(2)U’
(18gU’ +20Uc3))y’]
+ 12U’g2+6U(5’)y’]},
+ 15Ut2’)y
(2.14)
(3 ) Method of Cash, Raptis and Simos [ 111: 347
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21 June 1993
LTEcRs = h *ERCRs , ER CRS=-~~{d2[(g2+9U(2))~+2Uly’]+d[(2g3+20U’2+30gU(2)+14U(4))y+(12gU’+16U(3))y’] + [ (g4+28U2g+22U(‘)g2+ + (24U’3’g+48U’2’U’
16V4’g+
U’6’+26U’3’U’
+ 12U’g2+6U’5’)y’]}
f 15V2’)y
.
(2.15)
To compare (2.9)-(2.12) and (2.13)-(2.15), as in ref. [ 121, we distinguish two situations in terms of the value of E. (i) The energy E is close to the potential. Then d= UC-EzO. So only the free terms of the polynomials in d are identical in these equations. Thus, for these values of E methods (2.9)-(2.12) are of comparable accuracy. We have the result for methods (2.13)-(2.15). (ii) d > 0 or d c 0. Then ( d 1 is a large number. So, we have the following asymptotic expansions of eqs. (2.9)-(2.12) and (2.13)-(2.15) respectively. Two-step methods
ER, = - &,d3y, ER,
(2.16)
= - &d2gy,
(2.17)
ER,, = -&,{y[d(2g2+5U’2’)+2dU’y’},
(2.18)
ER, = - &dV2’y.
(2.19)
Hybrid methods
E&R = - niimd4y,
(2.20)
ERRC = - iksd3gy,
(2.21)
ER oRS= -&d’[
(g2+9U’2’)y+2U’y’]
.
(2.22)
From the above equations we have that in relation to (a) the method of Numerov the error increases as a third power of d, (b) the method of Raptis and Allison as a second power of d, (c) the method of Ixaru and Rizea as a first power of d and (d) the method of Raptis as a first power of d with a smaller coefficient than the method of Ixaru and Rizea. So we have the following theorem. Theorem 1. The method of Raptis [4] is the most accurate exponential-fitted two-step method merical integration for the Schrodinger equation, especially for large values of 1dl = 1UC- El.
for the nu-
From the above equations, we also have that in relation to (a) the method of Cash and Raptis the error increases as a fourth power of d, (b) the method of Raptis and Cash as a third power of d, and (c) the method of Cash, Raptis and Simos as a second power of d. So we have the following theorem. Theorem 2. The method of Cash, Raptis and Simos [ 111 is the most accurate exponential-fitted hybrid method for the numerical integration of the Schriidinger equation, especially for large values of IdI = I UC-El.
348
Volume
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4,5
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A
2.2. Four-step methods The local truncation errors of the above mentioned ( 1) Method of Henrici [ 81: LTEH=-&$ (2) Method
h8yviii(x)
+O(h’O)
I of Raptis
four-step
methods
and hybrid methods
are given by (2.23)
,
[ 51: (2.24)
LTERI=-&h8[yviii(~)-+‘i(~)], (3) Method
II of Raptis
[6]:
(4) Method
(2.25)
,
LTERIl=-~h8[yviii(x)-3v2y”(x)+3v4y”(x)-v6yii(x)] of Simos [ 81:
(2.26) With similar calculations ( 1) Method of Henrici:
as in section 2.1 we have the following expressions.
LTEH = h 8ERr, , ERH=-&{d4y+4d3gy+d2[(6U2+22Uc2))y+12U’y’] +d[ (4Ut3++28U’ (2 ) Method
+44U”‘g+
16UC4’)y+24(
U’g+UC3’)y’]
(2.27)
+GF} ,
I of Raptis:
LTERI = h 8ERR1 ,
+d[ (3g3+24U’2+37U”2’g+ (3) Method
15U’4’)y+
( 18U’g+20U’3’)y’]
(2.28)
+GF) ,
II of Raptis:
LTERu = h 8ERR,I , ERRrI = -&{4d2U”‘y+d[ (4) Method
(g3+ 16U’2+23U’2’g+
13U’4’)y+
(6U’g+
12UC3))y’] +GF} ,
(2.29)
of Simos:
LTEs = h 8ERs , ER SC -&${d[
( 12V4’+
16gU”‘+
12U’2’)y+8U0’y’]
SGF} ,
(2.30)
where GF=(g4+28u’2g+22U’2’g2+16U’4’g+U’6’+26U(3)u’+15U(2))y + (24U’3’g+48U’2’U’
+ 12U’g2+6U’5’)y’
.
(2.31)
From the above equations we have that in relation to (a) the method of Henrici the error increases as a fourth power of d, (b) method I of Raptis as a third power of d, (c) method II of Raptis as a second power of d and (d) the method of Simos as a first power of d. So we have the following theorem. Theorem 3. The method of Simos [ 81 is the most accurate exponential-fitted four-step method merical integration of the Schrijdinger equation, especially for large values of (dl= I UC-El.
for the nu-
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Volume 177, number 45
PHYSICS LETTERS A
References [ 1 ] A. Raptis and A.C. Allison, Comput. Phys. Commun. 14 ( 1978) 1. (21 T. Lyche, Numer. Math. 19 (1972) 65. [3] L.Gr. Ixaru and M. Rizea, Comput. Phys. Commun. 19 (1980) 23. [4] A.D. Raptis, Computing 28 (1982) 373. [5] A.D. Raptis, Comput. Phys. Commun. 28 (1983) 427. [ 61 A.D. Raptis, Comput. Phys. Commun. 24 ( 198 I ) 1. [ 71 P. Henrici, Discrete variable methods in ordinary differential equations (Wiley, New York, 1962) [8] T.E. Simos, J. Comput. Appl. Math. 30 (1990) 25 I. [9] J.R. Cash and A.D. Raptis, Comput. Phys. Commun. 33 ( 1984) 299. [lo] A.D. Raptis and J.R. Cash, Comput. Phys. Commun. 44 ( 1987) 95. [ 111 J.R. Cash, A.D. Raptis and T.E. Simos, J. Comput. Phys. 9 1 ( 1990) 413. [ 121 L.Gr. lxaru and M. Rizea, Comput. Phys. Commun. 38 (1985) 329.
350
21 June 1993
PHYSICS
LETTERS
A
Error analysis of exponential-fitted methods for the numerical solution of the one-dimensional Schredinger equation T.E. Simos Informatics Laboratory, Agricultural University ofAthens, Iera Odos 75. Athens 11855, Greece
Received 14 July 1992; accepted for publication 2 1 April 1993 Communicated by D.D. Holm
In this paper an error analysis of two-step and four-step methods for the numerical integration of the one-dimensional Schrijdinger equation is developed. The conclusion from this analysis is that the two-step Runge-Kutta type exponential fitted method by Cash, Raptis and Simos and the four-step exponential fitted method by Simos are among the now known exponential-fitted methods for the numerical solution of the radial Schriidinger equation the most effective ones.
1. Introduction There has always been a great activity from the computational point of view concerning the radial Schrodinger equation, the aim being to achieve a fast and reliable algorithm that generates a numerical solution. The radial Schrijdinger equation has the form
Y” (x)
=f(x)Y(x)
2
(1.1)
where f(x) = U(x) -E and U(x) =1(1+ 1 )/x2+ V(x) is an effective potential for which U(x)+0 as x+co. The boundary conditions are y(0) =O and another condition when x-*cr, which depends on the physical considerations. The most popular method for the numerical solution of ( 1.1) has been Numerov’s one. Raptis and Allison [ 11 based on the Chebyshevian multistep theory of Lyche [2] have developed a Numerov-type exponential fitted method and applied it to the Schrodinger equation corresponding to positive energies (i.e. solution in the asymptotic region). This method integrates exactly the polynomials 1, x, x2, exp( +vx). Based on this approach Ixaru and Rizea [ 31 have constructed an exponential-fitted method which integrates exactly the polynomials 1, x, exp( k vx), xexp( k VX). Also Raptis [ 41 has developed an exponential-fitted method which integrates exactly the polynomials exp ( k vx) , x exp ( 31ux) , x2 exp ( + vx) . We note that all these methods are fourthorder algebraic. Raptis [ 5,6] based on the classical method of Henrici [ 71 developed later a four-step sixth-order exponential-fitted method which integrates exactly the polynomials 1, x, x2, x3, x4, x5, exp ( ? vx) and 1, x, exp ( 2 vx), x exp ( f vx), x2 exp ( + vx) respectively and then Simos [ 8 ] developed an exponential-fitted method which integrates exactly the polynomials exp( 2 VX), x exp ( k vx), x2 exp( + VX), x3 exp( k vx). Cash and Raptis [ 93 developed a sixth-order hybrid method for the numerical solution of ( 1.1) which integrates exactly the polynomials 1, x, x2, x3, x4, x5, x6, x7. Raptis and Cash [lo] have developed on the other hand an exponential-fitted sixth-order hybrid method which integrates exactly the polynomials 1, x, x2, x3, x4, x5, exp( + vx). Finally Cash, Raptis and Simos [ 111 developed a sixth-order exponential-fitted method which integrates exactly the polynomials 1, x, x2, x3, exp( + VX), xexp( + VX). The purpose of this paper is to examine theoretically these methods and find a quantitative estimation for 03759601/93/$
06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
345
Volume 177, number 4,5
PHYSICS LETTERS A
21 June 1993
the extent of the accuracy gain to be expected from the optimal versions. We must note that an error analysis for three of the four-step methods and for three of the four two-step methods has been done by Ixaru and Rizea 13,121.
2. Error analysis 2.1. Two-step methods and hybrid methods The local truncation following.
errors of the above-mentioned
two-step methods
and hybrid
methods
is given by the
Two-step methods ( 1) Numerov’s
method:
(2.1)
LTE,.,=-&h6y”(x)+O(h8), (2) Method
[ 1] :
of Raptis and Allison
(2.2)
LTERA=-~h6[yvi(x)+~2yiv(~)]+O(h8), ( 3 ) Method
of Ixaru and Rizea
[ 3] :
LTE,R=-&,h6[y’i(x)-2u2yi’(x)+v4yii(x)]+O(h8), (4) Method
of Raptis
(2.3)
[4]:
LTEn = - &h6[yvi(x)-3v2yiV(x)+3v4yii(x)-zfj]+O(h*).
(2.4)
Hybrid methods ( 1) Method
of Cash and Raptis
LTEcR = - &h8yviii(x) (2 ) Method
[ 91:
+O(h’O)
,
of Raptis and Cash [ 10 ] :
LTERc = - ~oh8[yviii(x)-v2yyi(x)]+O(h10), (3) Method
(2.6)
of Cash, Raptis and Simos [ 111:
L=CRS = - ~h8[yviii(x)-3v2yVi(~)+3u4yiV(~)-v6yii(~)]+O(h’0). 120960 These formulae are still not suffkiently transparent to enable drawing quantitative conclusions of each method. To do this we applied them to the one-dimensional Schrijdinger equation. Based on the work of Ixaru and Rizea [ 12 1, we write first f(x) of ( 1.1) in the form f(x) =g(x)
(2.5)
+d 3
(2.7) on the merits
(2.8)
where g(x) = U(x) - U, =g, where U, is the constant approximation of the potential previously mentioned, and d= v2= UC-E. Now g(x) depends on the potential and its constant approximation and d stands for the energy dependence. Then we express the derivatives y”(x), y”‘(x), .... ye”‘(x), which are terms of the local error estimation for346
Volume
177, number
PHY SKIS LETTERS
4,5
21 June 1993
A
mula, in terms of the equation i.e. y” =f(x)y(x). We note that g(“) (x) = U(“) (x) for any nth order derivative with respect to x. We also express the terms as polynomials of d. Finally, we introduce these expressions in the local error estimation formulae (2.1)-( 2.7 ) and we obtain the following expressions. Two-step methods
( 1) Numerov’s method: LTEN = h 6ERN , ,
ER ,=-~~{y[d3+3d2g+d(3g2+7U(2))+g3+4U’2+7gU(2)+U(4)]+y’(6dU’+6gU’+4U(3)))
(2.9)
(2) Method of Raptis and Allison [ 11:
LTERA= h6ER,
,
(2.10)
ERRA=-~{y[d2g+d(2g2+6U(2))+g3+4U’2+7gU(2)+U(4)]+y’(4dU’+6gU’+4U(3))},
(3) Method of Ixaru and Rizea [ 31: LTEIR = h 6ER,R , ERIR =-~{y[d(2g2+5U(2’)+g3+4U’2+7gU’2’+U(4’]+y’(2dU’+6gIT’+4U(3))}
,
(2.11)
(4) Method of Raptis [4]:
LTER = h 6ERR , (2.12)
ER R=-&{y[4dU(2)+g3+4U’2+7gU(2)+U(4)]+y’(6gU’+4U0))}. Hybrid methods
( 1) Method of Cash and Raptis [ 91: LTEcR = h ‘ERCR , ERcR = - &{d4y+4d3gy+d2[ +d[ (4U’3’+28U’
(6U2+22Uc2’)y+
+44Uc2’g+
+ [ (g4+28U’2g+22U(2)g2+
+ (24U’3’g+48U’2’U’
16Ut4’)y+24(
12U’y’]
U’g+ Uc3))y’]
16V4’g+ U’6’+26U’3’U’ + 15U’2’)y
+ 12U’g2+6U’5’)y’]},
(2.13)
(2) Method of Raptis and Cash [ 10 1: LTERc = h *ERRC, ERRc = - &{d3gy+d2[
(3g2+ 15U’2’)y+6U’y’]
+d[ (3g3+24U’2+37gU’2’+
15U’4’)y+
+ [ (g4+28U’2g+22U(2)g2+
16Uc4’g+ U’6’+26U’3’U’
+ (24U’3’g+48U(2)U’
(18gU’ +20Uc3))y’]
+ 12U’g2+6U(5’)y’]},
+ 15Ut2’)y
(2.14)
(3 ) Method of Cash, Raptis and Simos [ 111: 347
Volume
177, number
4,5
PHYSICS
LETTERS
A
21 June 1993
LTEcRs = h *ERCRs , ER CRS=-~~{d2[(g2+9U(2))~+2Uly’]+d[(2g3+20U’2+30gU(2)+14U(4))y+(12gU’+16U(3))y’] + [ (g4+28U2g+22U(‘)g2+ + (24U’3’g+48U’2’U’
16V4’g+
U’6’+26U’3’U’
+ 12U’g2+6U’5’)y’]}
f 15V2’)y
.
(2.15)
To compare (2.9)-(2.12) and (2.13)-(2.15), as in ref. [ 121, we distinguish two situations in terms of the value of E. (i) The energy E is close to the potential. Then d= UC-EzO. So only the free terms of the polynomials in d are identical in these equations. Thus, for these values of E methods (2.9)-(2.12) are of comparable accuracy. We have the result for methods (2.13)-(2.15). (ii) d > 0 or d c 0. Then ( d 1 is a large number. So, we have the following asymptotic expansions of eqs. (2.9)-(2.12) and (2.13)-(2.15) respectively. Two-step methods
ER, = - &,d3y, ER,
(2.16)
= - &d2gy,
(2.17)
ER,, = -&,{y[d(2g2+5U’2’)+2dU’y’},
(2.18)
ER, = - &dV2’y.
(2.19)
Hybrid methods
E&R = - niimd4y,
(2.20)
ERRC = - iksd3gy,
(2.21)
ER oRS= -&d’[
(g2+9U’2’)y+2U’y’]
.
(2.22)
From the above equations we have that in relation to (a) the method of Numerov the error increases as a third power of d, (b) the method of Raptis and Allison as a second power of d, (c) the method of Ixaru and Rizea as a first power of d and (d) the method of Raptis as a first power of d with a smaller coefficient than the method of Ixaru and Rizea. So we have the following theorem. Theorem 1. The method of Raptis [4] is the most accurate exponential-fitted two-step method merical integration for the Schrodinger equation, especially for large values of 1dl = 1UC- El.
for the nu-
From the above equations, we also have that in relation to (a) the method of Cash and Raptis the error increases as a fourth power of d, (b) the method of Raptis and Cash as a third power of d, and (c) the method of Cash, Raptis and Simos as a second power of d. So we have the following theorem. Theorem 2. The method of Cash, Raptis and Simos [ 111 is the most accurate exponential-fitted hybrid method for the numerical integration of the Schriidinger equation, especially for large values of IdI = I UC-El.
348
Volume
177, number
PHYSICS
4,5
LETTERS
21 June1993
A
2.2. Four-step methods The local truncation errors of the above mentioned ( 1) Method of Henrici [ 81: LTEH=-&$ (2) Method
h8yviii(x)
+O(h’O)
I of Raptis
four-step
methods
and hybrid methods
are given by (2.23)
,
[ 51: (2.24)
LTERI=-&h8[yviii(~)-+‘i(~)], (3) Method
II of Raptis
[6]:
(4) Method
(2.25)
,
LTERIl=-~h8[yviii(x)-3v2y”(x)+3v4y”(x)-v6yii(x)] of Simos [ 81:
(2.26) With similar calculations ( 1) Method of Henrici:
as in section 2.1 we have the following expressions.
LTEH = h 8ERr, , ERH=-&{d4y+4d3gy+d2[(6U2+22Uc2))y+12U’y’] +d[ (4Ut3++28U’ (2 ) Method
+44U”‘g+
16UC4’)y+24(
U’g+UC3’)y’]
(2.27)
+GF} ,
I of Raptis:
LTERI = h 8ERR1 ,
+d[ (3g3+24U’2+37U”2’g+ (3) Method
15U’4’)y+
( 18U’g+20U’3’)y’]
(2.28)
+GF) ,
II of Raptis:
LTERu = h 8ERR,I , ERRrI = -&{4d2U”‘y+d[ (4) Method
(g3+ 16U’2+23U’2’g+
13U’4’)y+
(6U’g+
12UC3))y’] +GF} ,
(2.29)
of Simos:
LTEs = h 8ERs , ER SC -&${d[
( 12V4’+
16gU”‘+
12U’2’)y+8U0’y’]
SGF} ,
(2.30)
where GF=(g4+28u’2g+22U’2’g2+16U’4’g+U’6’+26U(3)u’+15U(2))y + (24U’3’g+48U’2’U’
+ 12U’g2+6U’5’)y’
.
(2.31)
From the above equations we have that in relation to (a) the method of Henrici the error increases as a fourth power of d, (b) method I of Raptis as a third power of d, (c) method II of Raptis as a second power of d and (d) the method of Simos as a first power of d. So we have the following theorem. Theorem 3. The method of Simos [ 81 is the most accurate exponential-fitted four-step method merical integration of the Schrijdinger equation, especially for large values of (dl= I UC-El.
for the nu-
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Volume 177, number 45
PHYSICS LETTERS A
References [ 1 ] A. Raptis and A.C. Allison, Comput. Phys. Commun. 14 ( 1978) 1. (21 T. Lyche, Numer. Math. 19 (1972) 65. [3] L.Gr. Ixaru and M. Rizea, Comput. Phys. Commun. 19 (1980) 23. [4] A.D. Raptis, Computing 28 (1982) 373. [5] A.D. Raptis, Comput. Phys. Commun. 28 (1983) 427. [ 61 A.D. Raptis, Comput. Phys. Commun. 24 ( 198 I ) 1. [ 71 P. Henrici, Discrete variable methods in ordinary differential equations (Wiley, New York, 1962) [8] T.E. Simos, J. Comput. Appl. Math. 30 (1990) 25 I. [9] J.R. Cash and A.D. Raptis, Comput. Phys. Commun. 33 ( 1984) 299. [lo] A.D. Raptis and J.R. Cash, Comput. Phys. Commun. 44 ( 1987) 95. [ 111 J.R. Cash, A.D. Raptis and T.E. Simos, J. Comput. Phys. 9 1 ( 1990) 413. [ 121 L.Gr. lxaru and M. Rizea, Comput. Phys. Commun. 38 (1985) 329.
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21 June 1993