Resolving mixing in Smoothed Particle Hydrodynamics
  Standard formulations of smoothed particle hydrodynamics (SPH) are unable to
resolve mixing at fluid boundaries. We use an error and stability analysis of
the generalised SPH equations of motion to prove that this is due to two
distinct problems. The first is a leading order error in the momentum equation.
This should decrease with increasing neighbour number, but does not because
numerical instabilities cause the kernel to be irregularly sampled. We identify
two important instabilities: the clumping instability and the banding
instability, and we show that both are cured by a suitable choice of kernel.
The second problem is the local mixing instability (LMI). This occurs as
particles attempt to mix on the kernel scale, but are unable to due to entropy
conservation. The result is a pressure discontinuity at boundaries that pushes
fluids of different entropy apart. We cure the LMI by using a weighted density
estimate that ensures that pressures are single valued throughout the flow.
This also gives a better volume estimate for the particles, reducing errors in
the continuity and momentum equations. We demonstrate mixing in our new
Optimised Smoothed Particle Hydrodynamics (OSPH) scheme using a Kelvin
Helmholtz instability (KHI) test with density contrast 1:2, and the 'blob test'
- a 1:10 density ratio gas sphere in a wind tunnel - finding excellent
agreement between OSPH and Eulerian codes.
