A New Sticky Particle Method for Pressureless Gas Dynamics

Abstract

We first present a new sticky particle method for the system of pressureless gas dynamics. The method is based on the idea of sticky particles, which seems to work perfectly well for the models with point mass concentrations and strong singularity formations. In this method, the solution is sought in the form of a linear combination of $\delta$-functions, whose positions and coefficients represent locations, masses, and momenta of the particles, respectively. The locations of the particles are then evolved in time according to a system of ODEs, obtained from a weak formulation of the system of PDEs. The particle velocities are approximated in a special way using global conservative piecewise polynomial reconstruction technique over an auxiliary Cartesian mesh. This velocities correction procedure leads to a desired interaction between the particles and hence to clustering of particles at the singularities followed by the merger of the clustered particles into a new particle located at their center of mass. The proposed sticky particle method is then analytically studied. We show that our particle approximation satisfies the original system of pressureless gas dynamics in a weak sense, but only within a certain residual, which is rigorously estimated. We also explain why the relevant errors should diminish as the total number of particles increases. Finally, we numerically test our new sticky particle method on a variety of one- and two-dimensional problems as well as compare the obtained results with those computed by a high-resolution finite-volume scheme. Our simulations demonstrate the superiority of the results obtained by the sticky particle method that accurately tracks the evolution of developing discontinuities and does not smear the developing $\delta$-shocks.