The use of compact boundary value method for the solution of two-dimensional Schrödinger equation.

*(English)*Zbl 1159.65081Summary: A high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing the spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods.

The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.

The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.

##### MSC:

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

35Q40 | PDEs in connection with quantum mechanics |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65L05 | Numerical methods for initial value problems |

##### Keywords:

Schrödinger equation; compact finite difference scheme in space; boundary value method in time; high accuracy; semidiscretization; stability; numerical experiments
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\textit{A. Mohebbi} and \textit{M. Dehghan}, J. Comput. Appl. Math. 225, No. 1, 124--134 (2009; Zbl 1159.65081)

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