Fourth-Order Algorithms for Solving the Imaginary Time Gross-Pitaevskii Equation in a Rotating Anisotropic Trap

TL;DR

This paper introduces highly efficient fourth-order algorithms for solving the imaginary time Gross-Pitaevskii equation in rotating anisotropic traps, significantly improving convergence speed over traditional methods.

Contribution

The authors develop a new class of fourth-order algorithms using forward, positive time step factorization schemes for nonlinear equations like the Gross-Pitaevskii equation.

Findings

Fourth-order algorithms converge at larger time steps than second-order methods.

The algorithms are highly accurate and fast for rotating anisotropic harmonic traps.

Time-dependent factorization schemes offer a systematic approach for nonlinear equation solutions.

Abstract

By implementing the exact density matrix for the rotating anisotropic harmonic trap, we derive a class of very fast and accurate fourth order algorithms for evolving the Gross-Pitaevskii equation in imaginary time. Such fourth order algorithms are possible only with the use of {\it forward}, positive time step factorization schemes. These fourth order algorithms converge at time-step sizes an order-of-magnitude larger than conventional second order algorithms. Our use of time-dependent factorization schemes provides a systematic way of devising algorithms for solving this type of nonlinear equations.