The Kelvin-Helmholtz instability and smoothed particle hydrodynamics
  We perform simulations of the Kelvin-Helmholtz instability using smoothed
particle hydrodynamics (SPH). The instability is studied both in the linear and
strongly non-linear regimes. The smooth, well-posed initial conditions of
Lecoanet et al. (2016) are used, along with an explicit Navier-Stokes viscosity
and thermal conductivity to enforce the evolution in the non-linear regime. We
demonstrate convergence to the reference solution using SPH. The evolution of
the vortex structures and the degree of mixing, as measured by a passive scalar
`colour' field, match the reference solution. Tests with an initial density
contrast produce the correct qualitative behaviour. The L2 error of the SPH
calculations decreases as the resolution is increased. The primary source of
error is numerical dissipation arising from artificial viscosity, and tests
with reduced artificial viscosity have reduced L2 error. A high-order smoothing
kernel is needed in order to resolve the initial velocity amplitude of the
seeded mode and inhibit excitation of spurious modes. We find that standard SPH
with an artificial viscosity has no difficulty in correctly modelling the
Kelvin-Helmholtz instability and yields convergent solutions.
