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Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation. (English) Zbl 1229.81011
Summary: In this paper, two compact finite difference schemes are presented for the numerical solution of the one-dimensional nonlinear Schrödinger equation. The discrete L 2 -norm error estimates show that convergence rates of the present schemes are of order O(h 4 +τ 2 ). Numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy.
MSC:
81-08Computational methods (quantum theory)
81Q05Closed and approximate solutions to quantum-mechanical equations
References:
[1]Davydov, A. S.: Solitons in molecular systems, (1985)
[2]Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C.: Solitons and nonlinear wave equations, (1982)
[3]Hasegawa, A.: Optical solitons in fibers, (1989)
[4]Sulem, C.; Sulem, P. L.: The nonlinear Schrödinger equation self-focusing and wave collapse, (1999)
[5]Ablowitz, M. J.; Segur, H.: Solitons and the inverse scattering transform, (1981)
[6]Taha, T. R.: A numerical scheme for the nonlinear Schrödinger equation, Comput. math. Appl. 22, No. 9, 77-84 (1991) · Zbl 0755.65130 · doi:10.1016/0898-1221(91)90208-L
[7]Taha, T. R.: Inverse scattering transform numerical schemes for nonlinear evolution equations and the method of lines, Appl. numer. Math. 20, 181-187 (1996) · Zbl 0940.65099 · doi:10.1016/0168-9274(95)00125-5
[8]Chang, Q.: Conservative scheme for a class of nonlinear Schrödinger equation, Chin. sci. Bull. 18, 1094-1097 (1981)
[9]Chang, Q.; Jia, E.; Sun, W.: Difference schemes for solving the generalized nonlinear Schrödinger equation, J. comput. Phys. 148, 397-415 (1999) · Zbl 0923.65059 · doi:10.1006/jcph.1998.6120
[10]Delfour, M.; Fortin, M.; Payre, G.: Finite difference solution of a nonlinear Schrödinger equation, J. comput. Phys. 44, 277-288 (1981) · Zbl 0477.65086 · doi:10.1016/0021-9991(81)90052-8
[11]Sanz-Serna, J. M.: Methods for the numerical solution of the nonlinear Schrödinger equation, Math. comput. 43, 21-27 (1984) · Zbl 0555.65061 · doi:10.2307/2007397
[12]Taha, T. R.; Ablowitz, M. J.: Analytical and numerical aspects of certain nonlinear evolution equations: II. Numerical nonlinear Schrödinger equation, J. comput. Phys. 55, 203-230 (1984) · Zbl 0541.65082 · doi:10.1016/0021-9991(84)90003-2
[13]Twizell, E. H.; Bratsos, A. G.; Newby, J. C.: A finite-difference method for solving the cubic Schrödinger equation, Math. comput. Simul. 43, 67-75 (1997) · Zbl 0886.65101 · doi:10.1016/S0378-4754(96)00056-0
[14]Wu, L.: Dufort – frankel-type methods for linear and nonlinear Schrödinger equations, SIAM J. Numer. anal. 33, 1526-1533 (1996) · Zbl 0860.65102 · doi:10.1137/S0036142994270636
[15]Zhang, L.: A high accurate and conservative finite difference scheme for nonlinear Schrödinger equation, Acta math. Appl. sin. 28, 178-186 (2005)
[16]Zhang, F.; Pérez-Grarciz, V. M.; Vázquez, L.: Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. math. Comput. 71, 165-177 (1995) · Zbl 0832.65136 · doi:10.1016/0096-3003(94)00152-T
[17]Argyris, J.; Haase, M.: An engineer’s guide to solitons phenomena: application of the finite element method, Comput. methods appl. Mech. engrg. 61, 71-122 (1987) · Zbl 0624.76020 · doi:10.1016/0045-7825(87)90117-4
[18]Gardner, L. R. T.; Gardner, G. A.; Zaki, S. I.; El Sahrawi, Z.: B-spline finite element studies of the non-linear Schrödinger equation, Comput. methods appl. Mech. engrg. 108, 303-318 (1993) · Zbl 0842.65083 · doi:10.1016/0045-7825(93)90007-K
[19]Herbst, B. M.; Morris, J. L.; Mitchell, A. R.: Numerical experience with the nonlinear Schrödinger equation, J. comput. Phys. 60, 282-305 (1985) · Zbl 0589.65084 · doi:10.1016/0021-9991(85)90008-7
[20]Tourigny, Y.; Morris, J. Ll.: An investigation into the effect of product approximation in the numerical solution of the cubic nonlinear Schrödinger equation, J. comput. Phys. 76, 103-130 (1988) · Zbl 0641.65090 · doi:10.1016/0021-9991(88)90133-7
[21]Karakashian, O.; Makridakis, C.: A space – time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. comput. 67, 479-499 (1998) · Zbl 0896.65068 · doi:10.1090/S0025-5718-98-00946-6
[22]Xu, Y.; Shu, C. -W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. comput. Phys. 205, 72-97 (2005) · Zbl 1072.65130 · doi:10.1016/j.jcp.2004.11.001
[23]Pathria, D.; Morris, J. Ll.: Pseudo-spectral solution of nonlinear Schrödinger equations, J. comput. Phys. 87, 108-125 (1990) · Zbl 0691.65090 · doi:10.1016/0021-9991(90)90228-S
[24]Sulem, P. L.; Sulem, C.; Patera, A.: Numerical simulation of singular solutions to the two-dimensional cubic Schrödinger equation, Comm. pure appl. Math. 37, 755-778 (1984) · Zbl 0543.65081 · doi:10.1002/cpa.3160370603
[25]Ablowitz, M. J.; Prinari, B.; Trubatch, A. D.: Discrete and continuous nonlinear Schrödinger systems, Bull. (New series) am. Math. soc. 43, 127-132 (2005)
[26]Sanz-Serna, J. M.; Verwer, J. G.: Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Numer. anal. 6, 25-42 (1986) · Zbl 0593.65087 · doi:10.1093/imanum/6.1.25
[27]Chippada, S.; Dawson, C. N.; Martínez, M. L.; Wheeler, M. F.: Finite element approximations to the system of shallow water equations. Part II: Discrete time a priori error estimates, SIAM J. Numer. anal. 36, 226-250 (1999) · Zbl 0958.76040 · doi:10.1137/S0036142996314159
[28]Dawson, C. N.; Martínez, M. L.: A characteristic-Galerkin approximation to a system of shallow water equations, Numer. math. 86, 239-256 (2000) · Zbl 0984.76043 · doi:10.1007/s002110000151
[29]Durán, A.; Lopez-Marcos, M. A.: Conservative numerical methods for solitary wave interactions, J. phys. A 36, 7761-7770 (2003) · Zbl 1038.35091 · doi:10.1088/0305-4470/36/28/306
[30]Zakharov, V. E.; Shabat, A. B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet phys. JETP 34, 62-69 (1972)