Abstract
ABSTRACT Smoothed particle hydrodynamics (SPH) is a Lagrangian method for solving the fluid equations that is commonplace in astrophysics, prized for its natural adaptivity and stability. The choice of variable to smooth in SPH has been the topic of contention, with smoothed pressure (P-SPH) being introduced to reduce errors at contact discontinuities relative to smoothed density schemes. Smoothed pressure schemes produce excellent results in isolated hydrodynamics tests; in more complex situations however, especially when coupling to the ‘sub-grid’ physics and multiple time-stepping used in many state-of-the-art astrophysics simulations, these schemes produce large force errors that can easily evade detection as they do not manifest as energy non-conservation. Here, two scenarios are evaluated: the injection of energy into the fluid (common for stellar feedback) and radiative cooling. In the former scenario, force and energy conservation errors manifest (of the same order as the injected energy), and in the latter large force errors that change rapidly over a few time-steps lead to instability in the fluid (of the same order as the energy lost to cooling). Potential ways to remedy these issues are explored with solutions generally leading to large increases in computational cost. Schemes using a density-based formulation do not create these instabilities and as such it is recommended that they are preferred over pressure-based solutions when combined with an energy diffusion term to reduce errors at contact discontinuities.
Highlights
Over the past three decades, the inclusion of hydrodynamics in galaxy formation simulations has become commonplace (Hernquist & Katz 1989; Evrard, Summers & Davis 1994; Springel & Hernquist 2002; Springel 2005; Dolag et al 2009)
The rest of this paper is organized as follows: In Section 2, the Smoothed particle hydrodynamics (SPH) method is described, along with the density- and pressure-based schemes; in Section 3, the basics of a galaxy formation model are discussed in more detail; in Section 4, issues relating to injection of energy into pressure-based schemes are explored; in Section 5, the SPH equations of motion are discussed; in Section 6, the timeintegration schemes used in cosmological simulations are presented and issues with sub-grid cooling are explored; and in Section 7, it is concluded that while pressure-SPH schemes can introduce significant errors it is possible in some cases to use measures to remedy them
The results presented here are not necessarily tied to the model used, and are applicable to a wide range of current galaxy formation models that use pressure-based SPH schemes
Summary
Over the past three decades, the inclusion of hydrodynamics in (cosmological) galaxy formation simulations has become commonplace (Hernquist & Katz 1989; Evrard, Summers & Davis 1994; Springel & Hernquist 2002; Springel 2005; Dolag et al 2009). The rest of this paper is organized as follows: In Section 2, the SPH method is described, along with the density- and pressure-based schemes; in Section 3, the basics of a galaxy formation model are discussed in more detail; in Section 4, issues relating to injection of energy into pressure-based schemes are explored; in Section 5, the SPH equations of motion are discussed; in Section 6, the timeintegration schemes used in cosmological simulations are presented and issues with sub-grid cooling are explored; and, it is concluded that while pressure-SPH schemes can introduce significant errors it is possible in some cases to use measures (albeit computationally expensive ones) to remedy them Because of this added expense it is suggested that a density-based scheme is preferred, with an energy diffusion term used to mediate contact discontinuities
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