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A note on the numerical solution of the semilinear Schrödinger equation. (English) Zbl 1239.65053
Summary: The unique solvability of local and nonlocal boundary value problems for the semilinear Schrödinger equation in a Hilbert space is investigated. The convergence estimates for the solution of difference schemes are established. Some numerical examples illustrating the methods described in our work are given. Applicability of these methods to nonlinear Schrödinger equation is discussed.

65L12Finite difference methods for ODE (numerical methods)
Full Text: DOI
[1] M.E. Mayfield, Non-reflective boundary conditions for Schrödinger’s equation, Ph.D. Thesis, University of Rhode Island, 1989
[2] Gordeziani, D. G.; Avalishvili, G. A.: Time-nonlocal problems for Schrödinger type equations: I. Problems in abstract spaces. Differential equations 41, No. 5, 703-711 (2005) · Zbl 1081.35004
[3] Gordeziani, D. G.; Avalishvili, G. A.: Time-nonlocal problems for Schrödinger type equations: II. Results for specific problems. Differential equations 41, No. 6, 852-859 (2005) · Zbl 1082.35055
[4] Han, H.; Jin, J.; Wu, X.: A finite difference method for the one-dimensional Schrödinger equation on unbounded domain. Computers and mathematics with applications 50, 1345-1362 (2005) · Zbl 1092.65071
[5] Antoine, X.; Besse, C.; Mouysset, V.: Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions. Mathematics of computation 73, No. 248, 1779-1799 (2004) · Zbl 1053.65072
[6] Ashyralyev, A.; Piskarev, S.; Wei, S.: On well-posedness of the difference schemes for abstract parabolic equations in $Lp([0,1],E)$ spaces. Numerical functional analysis and optimization 23, No. 7--8, 669-693 (2002) · Zbl 1022.65095
[7] Ashyralyev, A.; Sirma, A.: Nonlocal boundary value problem for the Schrödinger equation. Computers and mathematics with applications 55, 392-407 (2008) · Zbl 1155.65368
[8] Ashyralyev, A.; Sirma, A.: A note on the modified Crank--Nicholson difference schemes for Schrödinger equation. Complex analysis and potential theory, Proceedings of the conference satellite to ICM 2006, 256-271 (2007) · Zbl 1144.65313
[9] Ashyralyev, A.; Yurtsever, A.: On a nonlocal boundary value problem for semilinear hyperbolic--parabolic equations. Nonlinear analysis 47, 3585-3592 (2001) · Zbl 1042.65536
[10] Ben Mabrouk, A.; Ayadi, M.: A linearized finite-difference method for the solution of some mixed concave and non-linear problems. Applied mathematics and computation 197, 1-10 (2008) · Zbl 1135.65357
[11] Xanthopoulos, P.; Zouraris, G. E.: A linearly implicit finite difference method for a Klein--Gordon--Schrödinger system modeling electron--ion plasma waves. Discrete and continuous dynamical systems series B 10, 239-263 (2008) · Zbl 1145.82024