A scalable consistent second-order SPH solver for unsteady low Reynolds number flows

Smoothed Particle Hydrodynamics (SPH) has successfully been used to study a variety of cases involving nearly inviscid flows where conservation properties allow for good physical approximation despite poor theoretical approximation properties of differential operators. When used to study unsteady low Reynolds number flow with large dissipation, conservation alone cannot ensure quality of approximation and the traditional approach is inconsistent. An alternative formulation has recently become popular making use of an approximate splitting scheme to ensure a divergence-free velocity field. However, this scheme relies on an inconsistent discretization of the Laplacian that diverges as particles become disordered under flow. We present an incremental pressure correction scheme and combination of existing differential operator renormalizations that are able to achieve second order accuracy in time and space. A brief review of SPH approximation theory is provided to highlight the necessity of these renormalizations in implementing an approximate factorization scheme. We demonstrate that when fast algebraic multigrid preconditioners are used to solve the resulting linear systems, the scheme results in a consistent approximation that is scalable and amenable to parallelization. Several validation cases are presented for which a speedup of several orders of magnitude is achieved over traditional SPH approaches. Finally, an interface has been developed between the particle library LAMMPS and the sparse linear algebra libraries in Trilinos providing a massively parallel 3D SPH capability. Scaling results for up to 134 million particles on 32,768 cores are presented along with a demonstration of the capability to simulate complex 3D geometries. These results show that the added complexity of applying the necessary consistency corrections actually provides a factor of four speed-up per linear solver iteration versus the uncorrected case, despite the additional cost of constructing the corrections. The resulting library provides a method that is consistent, efficient, and second order in both space and time while maintaining the flexibility of classical SPH for single phase flows.