Comparison of two different integration methods for the (1+1)-Dimensional Schr\"odinger-Poisson Equation

TL;DR

This paper compares two numerical methods, Strang splitting and B-spline basis functions, for solving the 1D Schr"odinger-Poisson system, demonstrating the advantages of the splitting method in convergence and efficiency.

Contribution

It provides a comparative analysis of two numerical integration techniques for the Schr"odinger-Poisson equation, highlighting the effectiveness of the Strang splitting scheme.

Findings

Strang splitting method shows better convergence and efficiency.

Adaptive time-stepping allows larger simulation boxes.

Results suggest potential extension to 2D for cosmological applications.

Abstract

We compare two different numerical methods to integrate in time spatially delocalized initial densities using the Schr\"odinger-Poisson equation system as the evolution law. The basic equation is a nonlinear Schr\"odinger equation with an auto-gravitating potential created by the wave function density itself. The latter is determined as a solution of Poisson's equation modelling, e.g., non-relativistic gravity. For reasons of complexity, we treat a one-dimensional version of the problem whose numerical integration is still challenging because of the extreme long-range forces (being constant in the asymptotic limit). Both of our methods, a Strang splitting scheme and a basis function approach using B-splines, are compared in numerical convergence and effectivity. Overall, our Strang-splitting evolution compares favourably with the B-spline method. In particular, by using an adaptive time-stepper rather large one-dimensional boxes can be treated. These results give hope for extensions to two spatial dimensions for not too small boxes and large evolution times necessary for describing, for instance, dark matter formation over cosmologically relevant scales.