Sinc Collocation Numerical Methods for Solving Two-Dimensional Gross-Pitaevskii Equations with Non-Homogeneous Dirichlet Boundary Conditions

Abstract

This paper presents the numerical solution of the time-dependent Gross-Pitaevskii Equation describing the movement of quantum mechanics particles under non-homogeneous boundary conditions. Due to their inherent non-linearity, the equation generally can not be solved analytically. Instead, a highly accurate approximation to the solutions defined in a finite domain is proposed, using the Crank-Nicolson difference method and Sinc Collocation numerical methods to discretize separately in time and space. Two Sinc numerical approaches, involving the Sinc Collocation Method (SCM) and the Sinc Derivative Collocation Method (SDCM), are easy to implement. The results demonstrate that Sinc numerical methods are highly efficient and yield accurate results. Mainly, the SDCM decays errors faster than the SCM. Future work suggests that the SDCM can be extensively applied to approximate solutions under other boundary conditions to demonstrate its broad applicability further.