Rapid convergence for simulations that project from particles onto a fixed mesh

The advantage of particle Lagrangian methods in computational fluid dynamics is that advection is accurately modeled. However, this complicates the calculation of space derivatives. If a mesh is employed, it must be updated at each time step. On the other hand, fixed mesh, Eulerian, formulations benefit from the mesh being defined at the beginning of the simulation, but feature non-linear advection terms. It therefore seems natural to combine the two approaches, using a fixed mesh to perform calculations related to space derivatives, and using the particles to advect the information with time. The idea of combining Lagrangian particles and a fixed mesh goes back to Particle-in-Cell methods, and is here considered within the context of the finite element method (FEM) for the fixed mesh, and the particle FEM (pFEM) for the particles. Our results, in agreement with recent works, show that interpolation ("projection") errors, especially from particles to mesh, are the culprits of slow convergence of the method if standard, linear, FEs, are employed. By including quadratically consistent shape functions for the particles, the rate of convergence is restored to values competitive with purely Lagrangian simulations. The procedure is validated on the Zalesak's disk problem, the Taylor-Green vortex sheet flow, and the Rayleigh-Taylor instability.