Efficient High-order Mass-conserving and Energy-balancing Schemes for Schr\"odinger-Poisson Equations

TL;DR

This paper introduces relaxation-based numerical schemes for Schr"odinger-Poisson systems that ensure mass and energy conservation or balance, applicable with various time-stepping methods and demonstrated through 3D cosmological simulations.

Contribution

It develops and analyzes relaxation techniques integrated with implicit-explicit Runge-Kutta schemes for conserving mass and energy in Schr"odinger-Poisson equations, including time-varying coefficient cases.

Findings

Fully-discrete systems conserve mass and energy up to rounding errors.

Methods are effective in numerical examples, including 3D cosmological simulations.

Abstract

We study relaxation-based approaches for conserving mass and energy in the numerical solution of Schr\"odinger-Poisson (SP) type systems. Relaxation-based methods offer a general approach that can be applied as post-time step processing to achieve conservation with any time-stepping scheme. Here we study two types of relaxation techniques applied to implicit-explicit Runge-Kutta schemes, with Fourier collocation in space. We also study SP equations with time-varying coefficients (which appear naturally in cosmology) where energy is not conserved but satisfies a balance equation. We show that the fully-discrete system conserves both mass and energy (or satisfies the balance equation in case of time-varying coefficients), up to rounding errors. The effectiveness of these methods is demonstrated via numerical examples, including a three-dimensional cosmological simulation.