Numerical solution of the non-linear Schrodinger equation using smoothed-particle hydrodynamics

TL;DR

This paper introduces a novel smoothed-particle hydrodynamics method adapted for solving the non-linear Schrödinger equation, enabling robust, adaptive simulations of quantum systems like Bose-Einstein condensates and dark matter halos.

Contribution

It develops a new discretization of quantum pressure within SPH, extending fluid dynamics techniques to quantum wave equations for the first time.

Findings

Successfully simulates Bose-Einstein condensates and dark matter halos

Demonstrates robustness and adaptivity in complex quantum scenarios

Preserves conservation laws in numerical solutions

Abstract

We formulate a smoothed-particle hydrodynamics numerical method, traditionally used for the Euler equations for fluid dynamics in the context of astrophysical simulations, to solve the non-linear Schrodinger equation in the Madelung formulation. The probability density of the wavefunction is discretized into moving particles, whose properties are smoothed by a kernel function. The traditional fluid pressure is replaced by a quantum pressure tensor, for which a novel, robust discretization is found. We demonstrate our numerical method on a variety of numerical test problems involving the simple harmonic oscillator, Bose-Einstein condensates, collapsing singularities, and dark matter halos governed by the Gross-Pitaevskii-Poisson equation. Our method is conservative, applicable to unbounded domains, and is automatically adaptive in its resolution, making it well suited to study problems with collapsing solutions.