The numerical treatment of the nonlinear Schrödinger equation.

*(English)*Zbl 0702.65096The author presents a finite difference method of fourth-order accuracy in space and second order accuracy in time for integration of the nonlinear Schrödinger equation which is used as a model equation in computational physics, partly because it models two of the basic processes in a physical system, namely dispersion and nonlinearity.

It is shown that the method is exactly conservative, analytically and numerically. Three increasingly difficult problems are tested by the proposed method. The results are presented with the help of 3 tables and 3 figures. It is mentioned in tables 1-3 that \(\| u\|^ 2\) is conserved. However, the calculations show that \(L^ 2\)-norm for table 2 corresponding to worked out problem 2 does have changes in 11th significant decimal. The author has taken three iterations as a maximum number for finding the approximate solutions at every time step, while J. M. Sanz-Serna and J. G. Verwer [IMA J. Numer. Anal. 6, 25- 42 (1986; Zbl 0593.65087)] had taken 20 iterations as a maximum for problem 2.

Numerical results, presented in the form of tables and graphics drawn, appear to show a good agreement with the exact solution and a marked improvement over some methods suggested by Sanz-Serna.

It is shown that the method is exactly conservative, analytically and numerically. Three increasingly difficult problems are tested by the proposed method. The results are presented with the help of 3 tables and 3 figures. It is mentioned in tables 1-3 that \(\| u\|^ 2\) is conserved. However, the calculations show that \(L^ 2\)-norm for table 2 corresponding to worked out problem 2 does have changes in 11th significant decimal. The author has taken three iterations as a maximum number for finding the approximate solutions at every time step, while J. M. Sanz-Serna and J. G. Verwer [IMA J. Numer. Anal. 6, 25- 42 (1986; Zbl 0593.65087)] had taken 20 iterations as a maximum for problem 2.

Numerical results, presented in the form of tables and graphics drawn, appear to show a good agreement with the exact solution and a marked improvement over some methods suggested by Sanz-Serna.

Reviewer: H.K.Verma

##### 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 |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

##### Keywords:

stability; comparison with other methods; finite difference method; nonlinear Schrödinger equation
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\textit{A. B. Shamardan}, Comput. Math. Appl. 19, No. 7, 67--73 (1990; Zbl 0702.65096)

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##### References:

[1] | Sanz-Serna, J.M.; Verwer, J.G., IMA jl numer. analysis, 6, 25-42, (1986) |

[2] | Miles, J.W., SIAM jl appl. math., 41, 227-230, (1981) |

[3] | Herbst, B.M.; Mitchell, A.R., Report NA/66, (1983), University of Dundee, (revised version) |

[4] | B. M. Herbst, J. L. Morris and A. R. Mitchell, J. Comput. Phys. (in press). |

[5] | Delfour, M.; Fortin, M.; Payne, G., J. comput. phys., 44, 277-288, (1981) |

[6] | Griffiths, D.F.; Mitchell, A.R.; Morris, J.Li., Comput. meth. appl. mech. engng, 45, 177-215, (1984) |

[7] | Sanz-Serna, J.M.; Manoranjan, V.S., J. comput. phys., 52, 273-289, (1983) |

[8] | Sanz-Serna, J.M., Math. comput., 43, 21-27, (1984) |

[9] | Verwer, J.G.; Sanz-Serna, J.M., Computing, 33, 297-313, (1984) |

[10] | Weideman, J.A.C.; Herbst, B.M., SIAM jl numer. analysis, 23, 485-507, (1986) |

[11] | Zakharov, V.E.; Shabat, A.B., Soviet phys. JETP, 34, 62-69, (1972) |

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