Kelvin-Helmholtz instabilities with Godunov smoothed particle hydrodynamics

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. 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. 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 form 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 does not 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.