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

Several finite difference schemes are discussed for solving the two-dimensional Schrodinger 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 Barakat and Clark type explicit formula. 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.

[1]  David J. Evans,et al.  A note on the numerical solution of the schrödinger equation , 1990, Int. J. Comput. Math..

[2]  R. F. Warming,et al.  The modified equation approach to the stability and accuracy analysis of finite-difference methods , 1974 .

[3]  G. Pinder,et al.  Numerical solution of partial differential equations in science and engineering , 1982 .

[4]  John A. Clark,et al.  On the Solution of the Diffusion Equations by Numerical Methods , 1966 .

[5]  Wei-Ping Huang,et al.  The finite-difference vector beam propagation method: analysis and assessment , 1992 .

[6]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[7]  Weiying Zheng,et al.  Numerical solutions of the Schrödinger equation for the ground lithium by the finite element method , 2004, Appl. Math. Comput..

[8]  Garry J. Tee,et al.  Integration of Equations of Parabolic Type by the Method of Nets. , 1966 .

[9]  Mehdi Dehghan,et al.  Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition , 1999 .

[10]  Mehdi Dehghan,et al.  Parameter determination in a partial differential equation from the overspecified data , 2005, Math. Comput. Model..

[11]  Curtis F. Gerald,et al.  APPLIED NUMERICAL ANALYSIS , 1972, The Mathematical Gazette.

[12]  Mehdi Dehghan.  Numerical solution of a parabolic equation with non-local boundary specifications , 2003, Appl. Math. Comput..

[13]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[14]  V. Rosenberg,et al.  Methods for the Numerical Solution of Partial Differential Equations , 1969 .

[15]  A. R. Gourlay,et al.  Modified Locally One Dimensional Methods for Parabolic Partial Differential Equations in Two Space Variables , 1973 .

[16]  L.Gr. Ixaru,et al.  Operations on oscillatory functions , 1997 .

[17]  J. Crank,et al.  A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type , 1947 .

[18]  Mehdi Dehghan,et al.  Bounds for solutions of a six-point partial-difference scheme , 2004 .

[19]  F Y Hajj,et al.  Solution of the Schrodinger equation in two and three dimensions , 1985 .

[20]  G. M. Birtwistle.  The explicit solution of the equation of heat conduction , 1968, Comput. J..

[21]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[22]  Graeme Fairweather,et al.  Improved forms of the alternating direction methods of Douglas, Peaceman, and Rachford for solving parabolic and elliptic equations , 1964 .

[23]  B. Datta.  Numerical Linear Algebra and Applications , 1995 .