Implicit velocity projection method for incompressible SPH fluids

Abstract

Enforcing incompressibility has been extensively studied in smoothed particle hydrodynamics (SPH) fluid simulation. A variety of methods have been developed to this end, particularly based on implicit formulations to ensure stability and efficiency, which results in iterative pressure solvers. However, these iterative solvers, typically adopting the projected relaxed Jacobi (PRJ) method for the constant density constraint, suffer from slow convergence, mainly due to under-relaxation required for stability. In this paper, we interpret the PRJ iteration as an explicit fixed-point update that projects velocity onto the constraint manifold, and propose its implicit counterpart, the implicit velocity projection method. Our method enables significant over-relaxation, which leads to fast convergence while maintaining robustness, thereby achieving substantial performance gains. Our comprehensive experiments, including studies on different solvers with varying relaxation parameters, timestep sizes, density-error tolerances, and scene scales, demonstrate the superiority of our method over standard SPH methods.