Discretization correction of general integral PSE Operators for particle methods

The general integral particle strength exchange (PSE) operators [J.D. Eldredge, A. Leonard, T. Colonius, J. Comput. Phys. 180 (2002) 686–709] approximate derivatives on scattered particle locations to any desired order of accuracy. Convergence is, however, limited to a certain range of resolutions. For high-resolution discretizations, the constant discretization error dominates and prevents further convergence. We discuss a consistent discretization correction framework for PSE operators that yields the desired rate of convergence for any resolution, both on uniform Cartesian and irregular particle distributions, as well as near boundaries. These discretization-corrected (DC) PSE operators also have no overlap condition, enabling the kernel width to become arbitrarily small for constant interparticle spacing. We show that, on uniform Cartesian particle distributions, this leads to a seamless transition between DC PSE operators and classical finite difference stencils. We further identify relationships between DC PSE operators and operators used in corrected smoothed particle hydrodynamics and reproducing kernel particle methods. We analyze the presented DC PSE operators with respect to accuracy, rate of convergence, computational efficiency, numerical dispersion, numerical diffusion, and stability.