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

Understand

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.