A high - order accurate particle - in - cell method

SUMMARY We propose the use of high-order weighted essentially non-oscillitory interpolation and moving-least-squares approximation schemes alongside high-order time integration to enable high-order accurate particle-in-cell methods. The key insight is to view the unstructured set of particles as the underlying representation of the continuous fields; the grid used to evaluate integro-differential coupling terms is purely auxiliary. We also include a novel regularization term to avoid the accumulation of noise in the particle samples without harming the convergence rate. We include numerical examples for several model problems: advection-diffusion, shallow water, and incompressible Navier-Stokes in vorticity formulation. The implementation demonstrates fourth-order convergence, shows very low numerical dissipation, and is competitive with high-order Eulerian schemes.