Open Access

ARTICLE

Solution of Two-dimensional Linear and Nonlinear Unsteady Schrödinger Equation using “Quantum Hydrodynamics” Formulation with a MLPG Collocation Method

V. C. Loukopoulos¹, G. C. Bourantas²

1 Department of Physics, University of Patras, Patras, 26500, Rion, Greece.
2 Faculty of Science, Technology and Communication, University of Luxembourg, Campus Kirchberg, 6, rue Richard Coudenhove-Kalergi L-1359, Luxembourg.

Computer Modeling in Engineering & Sciences 2014, 103(1), 49-70. https://doi.org/10.3970/cmes.2014.103.049

Abstract

A numerical solution of the linear and nonlinear time-dependent Schrödinger equation is obtained, using the strong form MLPG Collocation method. Schrödinger equation is replaced by a system of coupled partial differential equations in terms of particle density and velocity potential, by separating the real and imaginary parts of a general solution, called a quantum hydrodynamic (QHD) equation, which is formally analogous to the equations of irrotational motion in a classical fluid. The approximation of the field variables is obtained with the Moving Least Squares (MLS) approximation and the implicit Crank-Nicolson scheme is used for time discretization. For the two-dimensional nonlinear Schrödinger equation, the lagging of coefficients method has been utilized to eliminate the nonlinearity of the corresponding examined problem. A Type-I nodal distribution is used in order to provide convergence for the discrete Laplacian operator used at the governing equation. Numerical results are validated, comparing them with analytical and numerical solutions.

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Keywords

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