Computational Eulerian Hydrodynamics and Galilean Invariance

TL;DR

This paper demonstrates that Eulerian hydrodynamical simulations are indeed Galilean invariant in the limit of high resolution, with numerical diffusion effects diminishing as resolution increases, ensuring accurate modeling of fluid instabilities.

Contribution

The study clarifies that Eulerian methods converge to Galilean-invariant solutions when sufficiently resolved, countering previous claims of non-invariance due to numerical diffusion effects.

Findings

Eulerian methods converge to Galilean-invariant solutions at high resolution.

Numerical diffusion effects diminish with increased resolution.

Kelvin-Helmholtz instabilities develop correctly with sufficient resolution.

Abstract

Eulerian hydrodynamical simulations are a powerful and popular tool for modeling fluids in astrophysical systems. In this work, we critically examine recent claims that these methods violate Galilean invariance of the Euler equations. We demonstrate that Eulerian hydrodynamics methods do converge to a Galilean-invariant solution, provided a well-defined convergent solution exists. Specifically, we show that numerical diffusion, resulting from diffusion-like terms in the discretized hydrodynamical equations solved by Eulerian methods, accounts for the effects previously identified as evidence for the Galilean non-invariance of these methods. These velocity-dependent diffusive terms lead to different results for different bulk velocities when the spatial resolution of the simulation is kept fixed, but their effect becomes negligible as the resolution of the simulation is increased to obtain a converged solution. In particular, we find that Kelvin-Helmholtz instabilities develop properly in realistic Eulerian calculations regardless of the bulk velocity provided the problem is simulated with sufficient resolution (a factor of 2-4 increase compared to the case without bulk flows for realistic velocities). Our results reiterate that high-resolution Eulerian methods can perform well and obtain a convergent solution, even in the presence of highly supersonic bulk flows.