A numerical study of adaptive space and time discretisations for Gross–Pitaevskii equations

Abstract

As a basic principle, benefits of adaptive discretisations are an improved balance between required accuracy and efficiency as well as an enhancement of the reliability of numerical computations. In this work, the capacity of locally adaptive space and time discretisations for the numerical solution of low-dimensional nonlinear Schrödinger equations is investigated. The considered model equation is related to the time-dependent Gross–Pitaevskii equation arising in the description of Bose–Einstein condensates in dilute gases. The performance of the Fourier-pseudo spectral method constrained to uniform meshes versus the locally adaptive finite element method and of higher-order exponential operator splitting methods with variable time stepsizes is studied. Numerical experiments confirm that a local time stepsize control based on a posteriori local error estimators or embedded splitting pairs, respectively, is effective in different situations with an enhancement either in efficiency or reliability. As expected, adaptive time-splitting schemes combined with fast Fourier transform techniques are favourable regarding accuracy and efficiency when applied to Gross–Pitaevskii equations with a defocusing nonlinearity and a mildly varying regular solution. However, the numerical solution of nonlinear Schrödinger equations in the semi-classical regime becomes a demanding task. Due to the highly oscillatory and nonlinear nature of the problem, the spatial mesh size and the time increments need to be of the size of the decisive parameter 0<ε≪1, especially when it is desired to capture correctly the quantitative behaviour of the wave function itself. The required high resolution in space constricts the feasibility of numerical computations for both, the Fourier pseudo-spectral and the finite element method. Nevertheless, for smaller parameter values locally adaptive time discretisations facilitate to determine the time stepsizes sufficiently small in order that the numerical approximation captures correctly the behaviour of the analytical solution. Further illustrations for Gross–Pitaevskii equations with a focusing nonlinearity or a sharp Gaussian as initial condition, respectively, complement the numerical study.

Reliability of the time splitting Fourier method for singular solutions in quantum fluids

“…Given the low regularity of vortex solutions, we use centered second order finite differences. With the aim of increasing the spatial resolution around the vortex cores and keeping a reasonable degree of freedom, we employ a set of nonuniform grid points (see [23], for instance, for locally adaptive finite element discretizations).…”

INFFTM: Fast evaluation of 3d Fourier series in MATLAB with an application to quantum vortex reconnections

“…|ψ(x)| 2σ+2 dx and the Strang splitting method described above can still be applied without any modification. We notice that space discretizations which are not regular (see, for instance, [5] for nonuniform finite differences and [18] for finite elements) provide results that are difficult to compare with those obtained via pseudospectral approaches, which are available only on regular grids. INFFTM allows the evaluation at arbitrary rectilinear grids and sets of arbitrary points making the comparison of these results possible.…”

Efficient time integration methods for Gross-Pitaevskii equations with rotation term

“…In essence, our implementation of CFQM exponential integrators is based on this formulation, i.e., our algorithm calls subroutines Φ (A) j and Φ (B) j depending on the coefficients (a jk ) K k=1 and b j , which yield approximations to the solutions of the subequations with adjusted time increments α h and β h. Additional information on time-splitting Fourier spectral space discretisation methods for autonomous Gross-Pitaevskii equations is found for instance in [30,31], see also Section 5. Due to the presence of a trapping potential, the values of the macroscopic wave function tend to zero; hence, a suitable restriction of the space domain and the application of the Fourier or, alternatively, Sine spectral method is justified, see [31].…”