An adaptive SPH method for strong shocks

We propose an alternative SPH scheme to usual SPH Godunov-type methods for simulating supersonic compressible flows with sharp discontinuities. The method relies on an adaptive density kernel estimation (ADKE) algorithm, which allows the width of the kernel interpolant to vary locally in space and time so that the minimum necessary smoothing is applied in regions of low density. We have performed a von Neumann stability analysis of the SPH equations for an ideal gas and derived the corresponding dispersion relation in terms of the local width of the kernel. Solution of the dispersion relation in the short wavelength limit shows that stability is achieved for a wide range of the ADKE parameters. Application of the method to high Mach number shocks confirms the predictions of the linear analysis. Examples of the resolving power of the method are given for a set of difficult problems, involving the collision of two strong shocks, the strong shock-tube test, and the interaction of two blast waves.