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# 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
Department of Physics, University of Patras, Patras, 26500, Rion, Greece.
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.

### Keywords

MLPG Collocation Method, Moving Least Squares, Schrödinger Equation, Quantum Hydrodynamics.

Loukopoulos, V. C., Bourantas, G. C. (2014). Solution of Two-dimensional Linear and Nonlinear Unsteady Schrödinger Equation using “Quantum Hydrodynamics” Formulation with a MLPG Collocation Method. CMES-Computer Modeling in Engineering & Sciences, 103(1), 49–70.

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