Fourth-order algorithms for solving the imaginary-time Gross-Pitaevskii equation in a rotating anisotropic trap

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 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.

Figure 1

Comparing the convergence of first- and second-order algorithms in computing the chemical potential of the Gross-Pitaevskii equation in a rotating anisotropic trap. The lines are fitted curves to algorithm 1A, 1B0, and 2A0 to demonstrate the order of convergence of each algorithm. The instability of data points in algorithms 1B0 and 2A0 are removed by the inclusion of one self-consistent $W$-function iteration as indicated by 1BW and 2AW.

Figure 2

The convergence of fourth-order algorithms in computing the chemical potential of the Gross-Pitaevskii equation in a rotating trap. Algorithms 4A00 (solid diamonds) and 4A0W (circles) are unstable beyond $\Delta \tau \approx 0.3$ and 0.5, respectively. Algorithm 4AWW (solid circles) is stable and showed excellent fourth-order convergence.

Figure 3

The convergence of various algorithms in computing the ground state energy of the GP equation. Both first-order results showed near-identical quadratic convergence. The second-order result 2AW (asterisks) is fourth order and 4AWW (solid circles) is eighth order. Results for 4A0W (circles) cannot be fitted because instability sets in abruptly at $\Delta \tau \approx 0.5$. Results for 4A00 is similar with instability at $\Delta \tau \approx 0.3$ and is not shown.