Journal of Computational Physics 152, 584–607 (1999) Article ID jcph.1999.6246, available online at http://www.idealibrary.com on An SPH Projection Method Sharen J. Cummins ∗ and Murray Rudman† ∗ Department of Mathematics and Statistics, Monash University, Clayton, Victoria 3168, Australia; and †CSIRO Building Construction and Engineering, Highett, Victoria 3122, Australia E-mail: ∗ sharen@groucho.maths.monash.edu.au and †Murray.Rudman@dbce.csiro.au Received September 16, 1998; revised February 19, 1999 A new formulation is introduced for enforcing incompressibility in Smoothed Particle Hydrodynamics (SPH). The method uses a fractional step with the velocity field integrated forward in time without enforcing incompressibility. The resulting intermediate velocity field is then projected onto a divergence-free space by solving a pressure Poisson equation derived from an approximate pressure projection. Unlike earlier approaches used to simulate incompressible flows with SPH, the pressure is not a thermodynamic variable and the Courant condition is based only on fluid velocities and not on the speed of sound. Although larger time-steps can be used, the solution of the resulting elliptic pressure Poisson equation increases the total work per time-step. Efficiency comparisons show that the projection method has a significant potential to reduce the overall computational expense compared to weakly compressible SPH, particularly as the Reynolds number, Re, is increased. Simulations using this SPH projection technique show good agreement with finite-difference solutions for a vortex spin-down and Rayleigh–Taylor instability. The results, however, indicate that the use of an approximate projection to enforce incompressibility leads to error accumulation in the density field. c  1999 Academic Press 1. INTRODUCTION Smoothed particle hydrodynamics (SPH) is a fully Lagrangian, particle-based technique that has typically been used to simulate the motion of compressible fluids. It was originally developed for astrophysical applications [12, 20] but has since been extended to model a wide range of problems including multi-phase flows [23], deformation and impact problems [34], and heat conduction [8]. More recently it has been extended and used to simulate the motion of incompressible fluids [24, 27]. Incompressibility is approximated in [24, 27] by assuming a compressible fluid with a large sound speed—typically a Mach number of M ≈ 0.1 is used. This approach will be termed here “weakly compressible” SPH or “WCSPH” and results obtained using this approach have been acceptable for free surface and some low Reynolds number flows, although not for fully confined moderate and high 584 0021-9991/99 $30.00 Copyright c  1999 by Academic Press All rights of reproduction in any form reserved.