Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices.

*(English)*Zbl 1089.65085Summary: Several finite difference schemes are discussed for solving the two-dimensional Schrödinger equation with Dirichlet’s boundary conditions. We use three fully implicit finite difference schemes, two fully explicit finite difference techniques, an alternating direction implicit procedure and the explicit formula of H. Z. Barakat and J. A. Clark [On the solution of the diffusion equation by numerical methods. ASME J. Heat Transf. 88, 421 (1966)]. Theoretical and numerical comparisons between four families of methods are described.

The main advantage of the alternating direction implicit finite difference technique is that the bandwidth of the sets of equations is a fixed small number that depends only on the nature of the computational molecule. This allows the use of very efficient and very fast techniques for solving the resulting tridiagonal systems of linear algebraic equations. The unique advantage of the Barakat and Clark technique is that it is unconditionally stable and is explicit in nature. Numerical results are presented followed by concluding remarks.

The main advantage of the alternating direction implicit finite difference technique is that the bandwidth of the sets of equations is a fixed small number that depends only on the nature of the computational molecule. This allows the use of very efficient and very fast techniques for solving the resulting tridiagonal systems of linear algebraic equations. The unique advantage of the Barakat and Clark technique is that it is unconditionally stable and is explicit in nature. Numerical results are presented followed by concluding remarks.

##### MSC:

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

78A25 | Electromagnetic theory, general |

35Q60 | PDEs in connection with optics and electromagnetic theory |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

78M20 | Finite difference methods applied to problems in optics and electromagnetic theory |

##### Keywords:

Two-dimensional Schrödinger equation; Electro-magnetic waves; Finite difference methods; Dirichlet’s boundary conditions; Optoelectronic devices; numerical examples; stability; alternating direction implicit (ADI) method; numerical comparisons
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\textit{M. Dehghan}, Math. Comput. Simul. 71, No. 1, 16--30 (2006; Zbl 1089.65085)

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