Solution of Two-dimensional Linear and Nonlinear Unsteady Schrodinger Equation using "Quantum Hydrodynamics" Formulation with a MLPG Collocation Method

A numerical solution of the linear and nonlinear time-dependent Schrodinger equation is obtained, using the strong form MLPG Collocation method. Schrodinger 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 Schrodinger 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.