Strong Shocks with Smoothed Particle Hydrodynamics

Abstract

Godunov-type methods relying on Riemann solvers have performed spectacularly well on supersonic compressible flows with sharp discontinuities as in the case of strong shocks. In contrast, the method of standard smoothed particle hydrodynamics (SPH) has been known to give a rather poor description of strong shock phenomena. Here we focus on the one-dimensional Euler equations of gas dynamics and show that the accuracy and stability of standard SPH can be significantly improved near sharp discontinuities if the bandwidth (or smoothing length, h) of the interpolating kernel is calculated by means of an adaptive density estimation procedure. Unlike existing adaptive SPH formulations, this class of estimates introduces less broad kernels in regions where the density is low, implying that the minimum necessary smoothing is applied in these regions. The resolving power of the method is tested against the strong shock-tube problem and the interaction of two blast waves. The quality of the solutions is comparable to that obtained using Godunov-type schemes and, in general, superior to that obtained from Riemann-based SPH formulations.