Fully implicit time integration in truly incompressible SPH

In the last years, the Smoothed Particle Hydrodynamics (SPH) method was introduced in new fields of engineering applications where highly viscous flow like polymer flow or very small geometries, like in micro-flow, are involved. In most of these applications it was not possible to use realistic fluid parameters, larger simulation domains or higher resolution because of restriction of time step due to the viscous time step criterion of the explicit time integration scheme. The computational effort was too high even for highly scalable SPH codes. In this article, we present a first-order implicit time stepping scheme for truly incompressible SPH (ISPH) to eliminate the viscous time step criterion. We propose a consistent time stepping approach where both velocity and particle position are solved implicitly and compare the results to traditional semi-implicit ISPH and implicit schemes where only velocity is integrated implicitly. We study Poiseuille flow, Taylor–Green flow and Rayleigh–Taylor instability and compare results of the implicit schemes to ISPH. Energy conservation is investigated using Taylor–Green vortex to highlight differences between the approaches. We find that the fully implicit time integration scheme is numerically stable. Since no error estimation of accuracy is used, the larger time step size leads to deviations of the trajectories. In future work, higher-order time integration schemes as well as error estimators should be investigated.