Hyperbolic Divergence Cleaning for SPH

TL;DR

This paper introduces a divergence cleaning method for SPH that conserves energy and significantly reduces divergence errors in magnetic and velocity fields, enhancing stability and accuracy in simulations.

Contribution

The authors develop an energy-conserving SPH divergence cleaning scheme based on Dedner et al's method, improving stability and error reduction in magnetic and velocity field simulations.

Findings

Average divergence error reduced by an order of magnitude.

Divergence errors maintained below 1% in 3D applications.

Density errors halved in oscillating water drop simulations.

Abstract

We present SPH formulations of Dedner et al's hyperbolic/parabolic divergence cleaning scheme for magnetic and velocity fields. Our implementation preserves the conservation properties of SPH which is important for stability. This is achieved by deriving an energy term for the Psi field, and imposing energy conservation on the cleaning subsystem of equations. This necessitates use of conjugate operators for divB and gradPsi in the numerical equations. For both the magnetic and velocity fields, the average divergence error in the system is reduced by an order of magnitude with our cleaning algorithm. Divergence errors in SPMHD are maintained to < 1%, even for realistic 3D applications with a corresponding gain in numerical stability. Density errors for an oscillating elliptic water drop using weakly compressible SPH are reduced by a factor of two.