Efficient High-order Mass-conserving and Energy-balancing Schemes for Schrödinger-Poisson Equations
This work addresses the need for accurate and efficient numerical schemes in computational physics, particularly for applications like cosmology, but it is incremental as it builds on existing relaxation techniques.
The paper tackles the problem of conserving mass and energy in numerical solutions of Schrödinger-Poisson equations by developing relaxation-based methods applied to implicit-explicit Runge-Kutta schemes, achieving conservation up to rounding errors as demonstrated in a 3D cosmological simulation.
We study relaxation-based approaches for conserving mass and energy in the numerical solution of Schrödinger-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.