A validated nonlinear KelvinHelmholtz benchmark for numerical hydrodynamics
Abstract
The nonlinear evolution of the KelvinHelmholtz instability is a popular test for code verification. To date, most KelvinHelmholtz problems discussed in the literature are illposed: they do not converge to any single solution with increasing resolution. This precludes comparisons among different codes and severely limits the utility of the KelvinHelmholtz instability as a test problem. The lack of a reference solution has led various authors to assert the accuracy of their simulations based on ad hoc proxies, e.g. the existence of smallscale structures. This paper proposes wellposed twodimensional KelvinHelmholtz problems with smooth initial conditions and explicit diffusion. We show that in many cases numerical errors/noise can seed spurious smallscale structure in KelvinHelmholtz problems. We demonstrate convergence to a reference solution using both ATHENA, a Godunov code, and DEDALUS, a pseudospectral code. Problems with constant initial density throughout the domain are relatively straightforward for both codes. However, problems with an initial density jump (which are the norm in astrophysical systems) exhibit rich behaviour and are more computationally challenging. In the latter case, ATHENA simulations are prone to an instability of the inner rolledup vortex; this instability is seeded by gridscale errors introduced by the algorithm, and disappears as resolution increases. Both ATHENA and DEDALUS exhibit latetime chaos. Inviscid simulations are riddled with extremely vigorous secondary instabilities which induce more mixing than simulations with explicit diffusion. Our results highlight the importance of running wellposed test problems with demonstrated convergence to a reference solution. To facilitate future comparisons, we include as supplementary material the resolved, converged solutions to the KelvinHelmholtz problems in this paper in machinereadable form.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 February 2016
 DOI:
 10.1093/mnras/stv2564
 arXiv:
 arXiv:1509.03630
 Bibcode:
 2016MNRAS.455.4274L
 Keywords:

 hydrodynamics;
 instabilities;
 methods: numerical;
 Astrophysics  Instrumentation and Methods for Astrophysics;
 Physics  Fluid Dynamics
 EPrint:
 Reference solution snapshots can be found at http://w.astro.berkeley.edu/~lecoanet/data/