Sinc-Collocation Difference Methods for Solving the Gross-Pitaevskii Equation

Description

The time-dependent Gross-Pitaevskii Equation, describing the movement of parti-
cles in quantum mechanics, may not be solved analytically due to its inherent non-
linearity. Hence numerical methods are of importance to approximate the solution.
This study develops a discrete scheme in time and space to simulate the solution
defined in a finite domain by using the Crank-Nicolson difference method and Sinc
Collocation Methods (SCM), respectively. In theory and practice, the time discretiz-
ing system decays errors in the second-order of accuracy, and SCMs are decaying
errors exponentially. A new SCM with a unique boundary treatment is proposed
and compared with the original SCM and other similar numerical techniques in time
costs and numerical errors. As a result, the new SCM decays errors faster than the
original one. Also, to attain the same accuracy, the new SCM interpolates fewer
nodes than the original SCM, which saves computational costs. The new SCM is
capable of approximating partial differential equations under different boundary con-
ditions, which can be extensively applied in fitting theory.
Author Keywords: Crank-Nicolson difference method, Gross-Pitaevskii Equation, Sinc-Collocation methods