Smoothed Particle Hydrodynamics and Magnetohydrodynamics

TL;DR

This paper provides a comprehensive overview of Smoothed Particle Hydrodynamics and Magnetohydrodynamics, discussing theoretical foundations, practical issues, and recent developments, along with releasing a new SPH code for testing algorithms.

Contribution

It offers a detailed critique of common misconceptions, practical solutions for SPMHD issues, and introduces a new publicly available SPH code for research and testing.

Findings

Clarifies the derivation of SPH equations from density estimates

Identifies and discusses numerical instabilities in SPMHD

Provides practical solutions and a new code for SPMHD simulations

Abstract

This paper presents an overview and introduction to Smoothed Particle Hydrodynamics and Magnetohydrodynamics in theory and in practice. Firstly, we give a basic grounding in the fundamentals of SPH, showing how the equations of motion and energy can be self-consistently derived from the density estimate. We then show how to interpret these equations using the basic SPH interpolation formulae and highlight the subtle difference in approach between SPH and other particle methods. In doing so, we also critique several `urban myths' regarding SPH, in particular the idea that one can simply increase the `neighbour number' more slowly than the total number of particles in order to obtain convergence. We also discuss the origin of numerical instabilities such as the pairing and tensile instabilities. Finally, we give practical advice on how to resolve three of the main issues with SPMHD: removing the tensile instability, formulating dissipative terms for MHD shocks and enforcing the divergence constraint on the particles, and we give the current status of developments in this area. Accompanying the paper is the first public release of the NDSPMHD SPH code, a 1, 2 and 3 dimensional code designed as a testbed for SPH/SPMHD algorithms that can be used to test many of the ideas and used to run all of the numerical examples contained in the paper.