Methods for the numerical solution of the nonlinear Schroedinger equation
Author:
J. M. SanzSerna
Journal:
Math. Comp. 43 (1984), 2127
MSC:
Primary 65M10
MathSciNet review:
744922
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Abstract: Optimal rates of convergence are established for several fullydiscrete schemes for the numerical solution of the nonlinear Schroedinger equation. Both finite differences and finite elements are considered for the discretization in space, while the integration in time is treated either by the leapfrog technique or by a modified CrankNicolson procedure, which generalizes the one suggested by Delfour, Fortin and Payne and possesses two useful conserved quantities.
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 T. B. Benjamin & J. E. Feir, "The disintegration of wave trains in deep water. Part 1," J. Fluid. Mech., v. 27, 1967, pp. 417430.
 [3]
 M. Delfour, M. Fortin & G. Payne, "Finitedifference solution of a nonlinear Schroedinger equation," J. Comput. Phys., v. 44, 1981, pp. 277288. MR 645840 (83c:65195)
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 G. Fairweather, Finite Element Galerkin Methods for Differential Equations, Marcel Dekker, New York, 1978. MR 0495013 (58:13781)
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 [6]
 Kuo PenYu & J. M. SanzSerna, "Convergence of methods for the numerical solution of the Kortewegde Vries equation", IMA J. Numer. Anal., v. 1, 1981, pp. 215221. MR 616332 (82e:65092)
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 J. C. LopezMarcos, Integración Numérica de la Ecuación de Schroedinger no lineal, M.Sc. thesis, University of Valladolid, Spain, 1983.
 [8]
 P. D. Richtmyer & K. W. Morton, Difference Methods for InitialValue Problems, InterscienceWiley, New York, 1967.
 [9]
 J. M. SanzSerna & V. S. Manoranjan, "A method for the integration in time of certain partial differential equations," J. Comput. Phys., v. 52, 1983, pp. 273289. MR 725596 (85d:65047)
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 W. A. Strauss, "The nonlinear Schroedinger equation" in Contemporary Developments in Continuum Mechanics and Partial Differential Equations (G. M. de la Penha and L. A. Medeiros, Eds.), NorthHolland, New York, 1978, pp. 452465. MR 519654 (81i:35047)
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 W. Strauss & L. Vazquez, "Numerical solution of a nonlinear KleinGordon equation," J. Comput. Phys., v. 28, 1978, pp. 271278. MR 0503140 (58:19970)
 [12]
 M. F. Wheeler, "Apriori estimates for Galerkin approximation to parabolic partial differential equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 723759. MR 0351124 (50:3613)
 [13]
 H. C. Yuen & W. E. Ferguson, Jr., "Relationship between BenjaminFeir instability and recurrence in the nonlinear Schroedinger equation," Phys. Fluids, v. 21, 1978, pp. 12751278.
 [14]
 V. E. Zakharov & A. B. Shabat, "Exact theory of twodimensional selffocusing and onedimensional selfmodulation of waves in nonlinear media," Soviet Phys. JETP, v. 34, 1972, pp. 6269. MR 0406174 (53:9966)
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DOI:
http://dx.doi.org/10.1090/S0025571819840744922X
PII:
S 00255718(1984)0744922X
Article copyright:
© Copyright 1984
American Mathematical Society
