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Numerical simulation of the nonlinear Schrödinger equation
Mathematics and Computers North-Holland
NUMERICAL
in Simulation
SIMULATION
309
32 (1990) 309-312
OF THE NONLINEAR
SCHRijDINGER
EQUATION
Thiab R. TAHA Computer Science Department,
A proposed scheme mented and compared related to the inverse numerical schemes for
The University of Georgia, Athens, GA 30602, USA
for the numerical simulation of the Nonlinear Schrodinger (NLS) equation is impleto other known numerical methods. This proposed scheme is constructed by methods scattering transform (IST). Performance data are used to compare it with other known the NLS equation.
1. Introduction In 1976 Ablowitz and Ladik found a nonlinear partial difference to the IST) which has as limiting form the NLS equation [l] iq, =
a, +21q12q*
equation
(by methods
related
(1)
The difference equation can be used as a numerical scheme for the NLS equation. In 1984 Taha and Ablowitz implemented the scheme, and found that it generally proved to be faster than all utilized finite difference methods but somewhat slower than the finite Fourier methods [2]. In 1984 Taha and Ablowitz derived nonlinear partial difference equations, based on the IST, which have as limiting forms the Korteweg-de Vries (KdV) and the Modified Kortewegde Vries (MKdV) equations [3]. In 1984 and 1988 they implemented the IST schemes for the KdV and MKdV equations and compared them with other known numerical methods for such equations [4,5]. They found that the IST schemes for the above equations to be faster than both the finite difference and the finite Fourier methods they considered. These results, for the KdV and MKdV equations, motivated a search for an IST scheme for the NLS equation which will perform better than finite Fourier methods. In this paper, a proposed numerical scheme, constructed by methods related to the IST, for the NLS equation given in (1) is presented. The linear part of this scheme has a truncation error of order 0((At)2) + O((AX)~), while the linear part for the Ablowitz-Ladik scheme has a truncation error of order 0((At)2) + O((AX)~). This proposed scheme is implemented and compared to the schemes in [2]. It is found that this new scheme is much faster than both the finite difference, including the Ablowitz-Ladik scheme, and the finite Fourier methods for small amplitudes.
0378-4754/90/$03.50
0 1990 - Elsevier
Science Publishers
B.V. (North-Holland)
T.R. Taha / Numerical simulation of Schriidinger equation
310
2. The proposed numerical scheme for the NLS equation The proposed numerical scheme which is based on the IST which was derived in [ll] is + 4,“+l - 4: = { (9nm+2
q,“_&P-
+ (AxI2
(4;;;
+ q,“_+21)D’4’+
(4s
+
(q,“+1
q,“_fi’)D?+
+ q,“_*)k?
q,“A’o’- q,“+lD’o’}
q:+Iq,m++2lqZ+;*+ q,m++llq:+l*qZ+l+ q,m_+llqLqn”-++1’*
ii
+ +(q;L-yq;+‘”
+ q,“_Iq;*)
+4nm+1(4L9,m++ll*
+
+
c29,+1*>
q,m+++llqn”-+l’* + q,“+2q,m*
+ q;+lqnm_*l)
A’4’
+ qnm_+llq,m_**lq,m_ 2 + q,“_ Iq,m_+llq,m* + 4: +
(4:
9
*4Lz
+ c+xi+~)
( q,m-+l’qn”++ll* +
+ cl,“_ 14,“+*1+
+ _
1
5
qnm-+21q,m+l*
c-
24,” *
( qn”+l*qn”++ll + q,m*q;+J
9;+l
( q,m+l* 4;:;
Dl”’
+ q~“q”-1q,““*}A”
+ q,“*q,“_ 1) +
q,m4,“*q::;
(2)
where * implies a complex conjugate, and A(O) = $ia ,
A?=
&iu ,
D(4,= -&ia,
A!)=
-:,ia ,
D’_2)= $ia 3
Eq. (2) is consistent with the NLS equation (l), with the truncation error of order 0((At)2
+
(Ax)~)3. Numerical implementation
and comparisons with other methods
To implement the proposed scheme given in Eq. (2), it is efficient sweeping technique. Write the new time level equation as -q,“_+; + 169;:;
- (30 + c)q;+’
+ 16q,“,+,’ - 4;;;
= B,,
to use the following (3)
T.R. Taha / Numerical simulation of Schrijdinger equation
311
where c = 24i(Ax)‘,‘At,
(6 ( K 1
and
+
bw2 2
I
q,m+ qnm+y* [ 4,“4,m++21 + qnm+1qz; + 4LdY
+ 4L4nm+11
+(qnm_++ll +qz-l)[qrt+lllqnm+l + 4,“~14,“*1 + (&-y + d- 1)* [4nmq,m++ll + 2cx- I 2
+ dYY)* [4,“+‘4,“_1+ 2q,“,1dzl + (cl,“+1 + 4,miI;14nm+1*] +14::12(C21 + 4,"+2) + (4% + q,“+Jq,m4,“_* -
+Ic+l I’(qz2l+ d-2)) Wx,‘( (C+’ + q:)*[OC+~+ d-1q7+1] + M2(4nm+1 +c3 +14nm+112(4nm_l +4AY)j.
Eq. (3) is solved by a back and fourth new time level) 4, = a,qn+l
sweep method.
We seek an equation
(4
of the form (at the
+ b,q,+2 + c,
(5)
suitable for computing q explicitly by sweeping to the left. Substitute the values of qn_2 and qn_ 1 from (5) into (3) and compare the coefficients of the resulting equation with (5) one gets a,, = - (16 + b,_,(16 c, = (B, - ~,-~(16
- a._,))/d,,
b, = l/d,,
- an-2) + ~,-~)/d,,
d, = a,_,(16
- an_2) - bn_2 - (30 + c). (6)
To obtain the solution q,, first solve for a,, b,,, and c, by sweeping to the right, then use Eq. (5) to calculate q,. In order to calculate the a’s, b’s, c’s, and the q’s we use an iteration procedure
WUI. To obtain a comparative evaluation of the efficiency of the new scheme, it is compared with the following schemes [2]: (1) Ablowitz-Ladik scheme [2], (2) Pseudospectral method (Fornberg & Whitham) [7], (3) the split-step Fourier method (F. Tappert) [8]. Our approach for comparison is the same as the one used in [2]. One soliton solution is used as an initial condition. Periodic boundary conditions are employed.
4. Conclusions According to our numerical experiments the following conclusions can be drawn: 1. The proposed scheme given in Eq. (2), based on IST, proved to be faster than all of the methods considered for an amplitude equal to 1. This scheme is 1.2 times faster than the Tappert method and 4.6 times faster than the second order Ablowitz-Ladik method.
31i
T. R. Taha / Numerical simulation of Schrijdinger equation
2. For an amplitude equal to 2, the Ablowitz-Ladik local scheme proved to be faster than the other schemes, followed by the proposed scheme and then by Tappert’s method. It is worth noting that there are other strong numerical schemes for solving nonlinear evolution equations, including a scheme using finite element techniques introduced by Bona, Dougahs, and Karakashian [9], and an adaptive numerical scheme introduced by Sanz-Serna [lo]. In the future it would be useful to compare the IST schemes to these other schemes as well as to consider conditions more general than those yielding interacting solitary waves.
Acknowledgements The author would like to thank Mark Ablowitz for helpful discussions, and Robert Robinson for his careful reading of this paper. I would also like to thank my wife Eman for her patience and support. This research has been supported in part by the U.S. Army Research Office.
References [l] M. Ablowitz and J. Lad& A nonlinear difference scheme and inverse scattering, Stud. AppZ. Math. 55 (1976) 213. [2] T.R. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrijdinger equation, J. Comput. Phys. 55 (2) (1984) 203. [3] T.R. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical, J. Comput. Phys. 55 (2) (1984) 192. [4] T.R. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation, J. Comput. Phys. 55 (2) (1984) 231. [5] T.R. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. IV. Numerical, modified Korteweg-de Vries equation, J. Comput. Phys. 77 (2) (1988) 540. [6] M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981). [7] B. Fomberg and G.B. Whitam, A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. Roy. Sot. 289 (1978) 373. [8] R.H. Hardin and F.D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM-SIGNUM Fall Meeting, Austin, TX (October 1972) SIAM Rev. Chronicle 15 (1973) 423. [9] J. Bona, V. Dougalis and 0. Karakashian, Fully discrete Galerkin methods for the Korteweg-de Vries equation, Comput. Math. AppZ. 12A (7) (1986) 859. [lo] J.M. Sanz-Sema, An explicit finite-difference scheme with exact conservation properties, J. Comput. Phys. 47 (1982) 199. [ll] T.R. Taha, A new IST numerical scheme for the nonlinear Schriidinger equation, to appear in Proc. IMACS 1st Intemat. Conf. Computational Physics, Boulder, CO., June 1990.
NUMERICAL
in Simulation
SIMULATION
309
32 (1990) 309-312
OF THE NONLINEAR
SCHRijDINGER
EQUATION
Thiab R. TAHA Computer Science Department,
A proposed scheme mented and compared related to the inverse numerical schemes for
The University of Georgia, Athens, GA 30602, USA
for the numerical simulation of the Nonlinear Schrodinger (NLS) equation is impleto other known numerical methods. This proposed scheme is constructed by methods scattering transform (IST). Performance data are used to compare it with other known the NLS equation.
1. Introduction In 1976 Ablowitz and Ladik found a nonlinear partial difference to the IST) which has as limiting form the NLS equation [l] iq, =
a, +21q12q*
equation
(by methods
related
(1)
The difference equation can be used as a numerical scheme for the NLS equation. In 1984 Taha and Ablowitz implemented the scheme, and found that it generally proved to be faster than all utilized finite difference methods but somewhat slower than the finite Fourier methods [2]. In 1984 Taha and Ablowitz derived nonlinear partial difference equations, based on the IST, which have as limiting forms the Korteweg-de Vries (KdV) and the Modified Kortewegde Vries (MKdV) equations [3]. In 1984 and 1988 they implemented the IST schemes for the KdV and MKdV equations and compared them with other known numerical methods for such equations [4,5]. They found that the IST schemes for the above equations to be faster than both the finite difference and the finite Fourier methods they considered. These results, for the KdV and MKdV equations, motivated a search for an IST scheme for the NLS equation which will perform better than finite Fourier methods. In this paper, a proposed numerical scheme, constructed by methods related to the IST, for the NLS equation given in (1) is presented. The linear part of this scheme has a truncation error of order 0((At)2) + O((AX)~), while the linear part for the Ablowitz-Ladik scheme has a truncation error of order 0((At)2) + O((AX)~). This proposed scheme is implemented and compared to the schemes in [2]. It is found that this new scheme is much faster than both the finite difference, including the Ablowitz-Ladik scheme, and the finite Fourier methods for small amplitudes.
0378-4754/90/$03.50
0 1990 - Elsevier
Science Publishers
B.V. (North-Holland)
T.R. Taha / Numerical simulation of Schriidinger equation
310
2. The proposed numerical scheme for the NLS equation The proposed numerical scheme which is based on the IST which was derived in [ll] is + 4,“+l - 4: = { (9nm+2
q,“_&P-
+ (AxI2
(4;;;
+ q,“_+21)D’4’+
(4s
+
(q,“+1
q,“_fi’)D?+
+ q,“_*)k?
q,“A’o’- q,“+lD’o’}
q:+Iq,m++2lqZ+;*+ q,m++llq:+l*qZ+l+ q,m_+llqLqn”-++1’*
ii
+ +(q;L-yq;+‘”
+ q,“_Iq;*)
+4nm+1(4L9,m++ll*
+
+
c29,+1*>
q,m+++llqn”-+l’* + q,“+2q,m*
+ q;+lqnm_*l)
A’4’
+ qnm_+llq,m_**lq,m_ 2 + q,“_ Iq,m_+llq,m* + 4: +
(4:
9
*4Lz
+ c+xi+~)
( q,m-+l’qn”++ll* +
+ cl,“_ 14,“+*1+
+ _
1
5
qnm-+21q,m+l*
c-
24,” *
( qn”+l*qn”++ll + q,m*q;+J
9;+l
( q,m+l* 4;:;
Dl”’
+ q~“q”-1q,““*}A”
+ q,“*q,“_ 1) +
q,m4,“*q::;
(2)
where * implies a complex conjugate, and A(O) = $ia ,
A?=
&iu ,
D(4,= -&ia,
A!)=
-:,ia ,
D’_2)= $ia 3
Eq. (2) is consistent with the NLS equation (l), with the truncation error of order 0((At)2
+
(Ax)~)3. Numerical implementation
and comparisons with other methods
To implement the proposed scheme given in Eq. (2), it is efficient sweeping technique. Write the new time level equation as -q,“_+; + 169;:;
- (30 + c)q;+’
+ 16q,“,+,’ - 4;;;
= B,,
to use the following (3)
T.R. Taha / Numerical simulation of Schrijdinger equation
311
where c = 24i(Ax)‘,‘At,
(6 ( K 1
and
+
bw2 2
I
q,m+ qnm+y* [ 4,“4,m++21 + qnm+1qz; + 4LdY
+ 4L4nm+11
+(qnm_++ll +qz-l)[qrt+lllqnm+l + 4,“~14,“*1 + (&-y + d- 1)* [4nmq,m++ll + 2cx- I 2
+ dYY)* [4,“+‘4,“_1+ 2q,“,1dzl + (cl,“+1 + 4,miI;14nm+1*] +14::12(C21 + 4,"+2) + (4% + q,“+Jq,m4,“_* -
+Ic+l I’(qz2l+ d-2)) Wx,‘( (C+’ + q:)*[OC+~+ d-1q7+1] + M2(4nm+1 +c3 +14nm+112(4nm_l +4AY)j.
Eq. (3) is solved by a back and fourth new time level) 4, = a,qn+l
sweep method.
We seek an equation
(4
of the form (at the
+ b,q,+2 + c,
(5)
suitable for computing q explicitly by sweeping to the left. Substitute the values of qn_2 and qn_ 1 from (5) into (3) and compare the coefficients of the resulting equation with (5) one gets a,, = - (16 + b,_,(16 c, = (B, - ~,-~(16
- a._,))/d,,
b, = l/d,,
- an-2) + ~,-~)/d,,
d, = a,_,(16
- an_2) - bn_2 - (30 + c). (6)
To obtain the solution q,, first solve for a,, b,,, and c, by sweeping to the right, then use Eq. (5) to calculate q,. In order to calculate the a’s, b’s, c’s, and the q’s we use an iteration procedure
WUI. To obtain a comparative evaluation of the efficiency of the new scheme, it is compared with the following schemes [2]: (1) Ablowitz-Ladik scheme [2], (2) Pseudospectral method (Fornberg & Whitham) [7], (3) the split-step Fourier method (F. Tappert) [8]. Our approach for comparison is the same as the one used in [2]. One soliton solution is used as an initial condition. Periodic boundary conditions are employed.
4. Conclusions According to our numerical experiments the following conclusions can be drawn: 1. The proposed scheme given in Eq. (2), based on IST, proved to be faster than all of the methods considered for an amplitude equal to 1. This scheme is 1.2 times faster than the Tappert method and 4.6 times faster than the second order Ablowitz-Ladik method.
31i
T. R. Taha / Numerical simulation of Schrijdinger equation
2. For an amplitude equal to 2, the Ablowitz-Ladik local scheme proved to be faster than the other schemes, followed by the proposed scheme and then by Tappert’s method. It is worth noting that there are other strong numerical schemes for solving nonlinear evolution equations, including a scheme using finite element techniques introduced by Bona, Dougahs, and Karakashian [9], and an adaptive numerical scheme introduced by Sanz-Serna [lo]. In the future it would be useful to compare the IST schemes to these other schemes as well as to consider conditions more general than those yielding interacting solitary waves.
Acknowledgements The author would like to thank Mark Ablowitz for helpful discussions, and Robert Robinson for his careful reading of this paper. I would also like to thank my wife Eman for her patience and support. This research has been supported in part by the U.S. Army Research Office.
References [l] M. Ablowitz and J. Lad& A nonlinear difference scheme and inverse scattering, Stud. AppZ. Math. 55 (1976) 213. [2] T.R. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrijdinger equation, J. Comput. Phys. 55 (2) (1984) 203. [3] T.R. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical, J. Comput. Phys. 55 (2) (1984) 192. [4] T.R. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation, J. Comput. Phys. 55 (2) (1984) 231. [5] T.R. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. IV. Numerical, modified Korteweg-de Vries equation, J. Comput. Phys. 77 (2) (1988) 540. [6] M. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981). [7] B. Fomberg and G.B. Whitam, A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. Roy. Sot. 289 (1978) 373. [8] R.H. Hardin and F.D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM-SIGNUM Fall Meeting, Austin, TX (October 1972) SIAM Rev. Chronicle 15 (1973) 423. [9] J. Bona, V. Dougalis and 0. Karakashian, Fully discrete Galerkin methods for the Korteweg-de Vries equation, Comput. Math. AppZ. 12A (7) (1986) 859. [lo] J.M. Sanz-Sema, An explicit finite-difference scheme with exact conservation properties, J. Comput. Phys. 47 (1982) 199. [ll] T.R. Taha, A new IST numerical scheme for the nonlinear Schriidinger equation, to appear in Proc. IMACS 1st Intemat. Conf. Computational Physics, Boulder, CO., June 1990.