  Numerical simulations for the non-linear development of Kelvin-Helmholtz
instability in two different density layers have been performed with the
particle-based method (Godunov SPH) developed by Inutsuka (2002). The Godunov
SPH can describe the Kelvin-Helmholtz instability even with a high density
contrast, while the standard SPH shows the absence of the instability across a
density gradient (Agertz et al. 2007). The interaction of a dense blob with a
hot ambient medium has been performed also. The Godunov SPH describes the
formation and evolution of the fingers due to the combinations of
Rayleigh-Taylor, Richtmyer-Meshkov, and Kelvin-Helmholtz instabilities. The
blob test result coincides well with the results of the grid-based codes. An
inaccurate handling of a density gradient in the standard SPH has been pointed
out as the direct reason of the absence of the instabilities. An unphysical
force happens at the density gradient even in a pressure equilibrium, and
repulses particles from the initial density discontinuity. Therefore, the
initial perturbation damps, and a gap forms at the discontinuity. The
unphysical force has been studied in terms of the consistency of a numerical
scheme. Contrary to the standard SPH, the momentum equation of the Godunov SPH
doesnt use the particle approximation, and has been derived from the kernel
convolution or a new Lagrangian function. The new Lagrangian function used in
the Godunov SPH is more analogous to the real Lagrangian function for
continuum. The momentum equation of the Godunov SPH has much better linear
consistency, so the unphysical force is greatly reduced compared to the
standard SPH in a high density contrast.
