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On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. (English) Zbl 0739.65096
For approximating the solution of a weakly nonlinear Schrödinger equation \(L^ 2\) conservative schemes are studied. The schemes are based on a space-discretization by the Galerkin method (in \(H^ 1_ 0\)). For the time-discretization two Crank-Nicolson type methods are used. Existence, uniqueness and convergence of the approximate solution to the exact one are proved. Existence, uniqueness and convergence of the approximate solution to the exact one are proved. The Newton method of “inner” iterations for solving the system of complex nonlinear equations is discussed. Numerical results are given.

MSC:
65Z05 Applications to the sciences
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
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References:
[1] Brezis, H., Gallouet, T.: Nonlinear Schr?dinger evolution equations. Nonlinear Analysis4, 677-681 (1980) · Zbl 0451.35023
[2] Browder, F.E.: Existence and uniqueness theorems for solutions of nonlinear boundary value problems. In: Finn, R. (ed.) Applications of Nonlinear Partial Differential Equations. Proc. Symp. Appl. Math. v. 17, pp. 24-49. Providence: American Mathematical Society 1965
[3] Delfour, M., Fortin, M., Payre, G.: Finite-difference solutions of a non-linear Schr?dinger equation. J. Comput. Phys.44, 277-288 (1981) · Zbl 0477.65086
[4] Griffiths, D.F., Mitchell, A.R., Morris, J.Ll.: A numerical study of the nonlinear Schr?dinger equation. Comput. Methods Appl. Mech. Eng.45, 177-215 (1984) · Zbl 0555.65060
[5] Herbst, B.M., Morris, J.Ll., Mitchell, A.R.: Numerical experience with the nonlinear Schr?dinger equation. J. Comput. Phys.60, 282-305 (1985) · Zbl 0589.65084
[6] Le Mesurier, B., Papanicolaou, G., Sulem, C., Sulem, P.-L.: The focusing singularity of the nonlinear Schr?dinger equation. In: Crandall, M.G., Rabinowitz, P.H., Turner, R.E. (eds.) Directions in Partial Differential Equations, pp. 159-201. New York: Academic Press 1987 · Zbl 0659.35020
[7] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. New York Berlin Heidelberg: Springer 1983 · Zbl 0516.47023
[8] Rasmussen, J.J., Rypdal, K.: Blow-up in Nonlinear Schr?dinger equations?I. A general review. Phys. Scr.33, 481-497 (1986) · Zbl 1063.35545
[9] Sanz-Serna, J.M.: Methods for the numerical solution of the nonlinear Schr?dinger equation. Math. Comput.43, 21-27 (1984) · Zbl 0555.65061
[10] Sanz-Serna, J.M., Verwer, J.G.: Conservative and nonconservative schemes for the solution of the nonlinear Schr?dinger equation. IMA J. Numer. Anal.6, 25-42 (1986) · Zbl 0593.65087
[11] Strauss, W.A.: The Nonlinear Schr?dinger equation. In: de la Penha, G.M., Medeiros, L.A.J., (eds.) Contemporary Developments in Continuum Mechanics and Partial Differential Equations. pp. 452-465. New York: North-Holland 1978
[12] Strauss, W.A., Vazquez, L.: Numerical solution of a nonlinear Klein-Gordon equation. J. Comp. Phys.28, 271-278 (1978) · Zbl 0387.65076
[13] Sulem, P.L., Sulem, C., Patera, A.: Numerical simulation of singular solutions to the two-dimensional cubic Schr?dinger equation. Comm. Pure Appl. Math.37, 755-778 (1984) · Zbl 0543.65081
[14] Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equations II. Numerical, nonlinear Schr?dinger equation. J. Comput. Phys.55, 203-230 (1984) · Zbl 0541.65082
[15] Thom?e, V.: Galerkin Finite Element Methods for Parabolic Problems. Lecture Notes in Mathematics v. 1054. Berlin Heidelberg New York: Springer 1984 · Zbl 0531.73052
[16] Tourigny, Y., Morris, J.LI: An investigation into the effect of product approximation in the numerical solution of the cubic nonlinear Schr?dinger equation. J. Comput. Phys.76, 103-130 (1988) · Zbl 0641.65090
[17] Verwer, J.G., Sanz-Serna, J.M.: Convergence of method of lines approximations to partial differential equations. Computing33, 297-313 (1984) · Zbl 0546.65064
[18] Weideman, J.A.C., Herbst, B.M.: Split-step methods for the solution of the nonlinear Schr?dinger equation. SIAM J. Numer. Anal.23, 485-507 (1986) · Zbl 0597.76012
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