Abstract
The nonlinear evolution of the Kelvin-Helmholtz instability is a popular test
for code verification. To date, most Kelvin-Helmholtz problems discussed in the
literature are ill-posed: they do not converge to any single solution with
increasing resolution. This precludes comparisons among different codes and
severely limits the utility of the Kelvin-Helmholtz 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
small-scale structures. This paper proposes well-posed Kelvin-Helmholtz
problems with smooth initial conditions and explicit diffusion. We show that in
many cases numerical errors/noise can seed spurious small-scale structure in
Kelvin-Helmholtz problems. We demonstrate convergence to a reference solution
using both Athena, a Godunov code, and Dedalus, a pseudo-spectral 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 behavior and are
more computationally challenging. In the latter case, Athena simulations are
prone to an instability of the inner rolled-up vortex; this instability is
seeded by grid-scale errors introduced by the algorithm, and disappears as
resolution increases. Both Athena and Dedalus exhibit late-time 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 well-posed test problems with demonstrated
convergence to a reference solution. To facilitate future comparisons, we
include the resolved, converged solutions to the Kelvin-Helmholtz problems in
this paper in machine-readable form.
Description
[1509.03630] A Validated Nonlinear Kelvin-Helmholtz Benchmark for Numerical Hydrodynamics
Links and resources
Tags