A WELL-POSED KELVIN-HELMHOLTZ INSTABILITY TEST AND COMPARISON

Abstract

Recently, there has been a significant level of discussion of the correct\ntreatment of Kelvin-Helmholtz instability in the astrophysical community. This\ndiscussion relies largely on how the KHI test is posed and analyzed. We pose a\nstringent test of the initial growth of the instability. The goal is to provide\na rigorous methodology for verifying a code on two dimensional Kelvin-Helmholtz\ninstability. We ran the problem in the Pencil Code, Athena, Enzo, NDSPHMHD, and\nPhurbas. A strict comparison, judgment, or ranking, between codes is beyond the\nscope of this work, though this work provides the mathematical framework needed\nfor such a study. Nonetheless, how the test is posed circumvents the issues\nraised by tests starting from a sharp contact discontinuity yet it still shows\nthe poor performance of Smoothed Particle Hydrodynamics. We then comment on the\nconnection between this behavior to the underlying lack of zeroth-order\nconsistency in Smoothed Particle Hydrodynamics interpolation. We comment on the\ntendency of some methods, particularly those with very low numerical diffusion,\nto produce secondary Kelvin-Helmholtz billows on similar tests. Though the lack\nof a fixed, physical diffusive scale in the Euler equations lies at the root of\nthe issue, we suggest that in some methods an extra diffusion operator should\nbe used to damp the growth of instabilities arising from grid noise. This\nstatement applies particularly to moving-mesh tessellation codes, but also to\nfixed-grid Godunov schemes.\n