Linearly implicit and large time-stepping conservative exponential relaxation schemes for the nonlocal cubic Gross-Pitaevskii equation

Abstract

The nonlocal cubic Gross-Pitaevskii equation, in comparison to the cubic Gross-Pitaevskii equation, incorporates a nonlocal diffusion operator and can capture a wider range of practical phenomena. However, this nonlocal formulation significantly increases the computational expenses in numerical simulations, necessitating the development of efficient and accurate time integration schemes. This paper uses the relaxation method to present two linearly implicit conservative exponential schemes for the nonlocal cubic Gross-Pitaevskii equation. One proposed scheme can inherit the discrete energy while the other preserves the mass in the discrete scene. We first apply the Fourier pseudo-spectral method to the equation and derive a conservative semi-discrete system. Then, based on the ideas of the traditional relaxation method, adopting the exponential time difference method to approximate the system in time can lead to an energy-preserving exponential scheme. The mass-preserving scheme is derived by using the integral factor method to discretize the system in the temporal direction. The stability results of the constructed schemes are given. In addition, all schemes are linearly implicit and can be implemented efficiently with a large time step. Finally, numerical results show that both proposed methods are remarkably efficient and have better stability than the original relaxation scheme.