Discretization correction of general integral PSE Operators for particle methods

Abstract

The general integral particle strength exchange (PSE) operators [J.D. Eldredge, A. Leonard, and T. Colonius, J. Comput. Phys. 180, 686-709 (2002)] 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. (2002)] 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.

A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

“…This would then result in the minimum number of particles needed for these approximations to reach below a certain error level everywhere in the domain. The above operators for approximating derivatives are consistent on almost any particle distribution [39], except those for which the associated Vandermonde matrix V is not invertible (see Eq (A.3)), in which case we randomly displace or insert particles until V becomes invertible.…”

“…(2.3b). We obtain a consistent approximation of derivatives of f on arbitrary distributions of particles with varying core sizes using DC-PSE operators [39], which rely on solving a small 1 linear system of equations at each particle to determine the kernel weights. After interpolating the f p values from the old particles to the new ones, there are two ways the right-hand side of Eq.…”

“…This requires the zeroth-order moment of the DC-PSE kernels to vanish. With DC-PSE operators this is possible since the zeroth-order moment is a free parameter that can be used to tune the stability properties of the operators [39]. Setting the zeroth-order moment to zero and evaluating the operators at off-particle locations makes DC-PSE a particle-analog of derivative-reproducing kernel (DRK) Galerkin collocation methods [12,46,44], which are conceptually related to Moving Least-Squares (MLS) schemes [26,6].…”

“…Rather, they are conceptually related to the interpolation scheme used in Behrens's adaptive semi-Lagrangian method [5] for radial basis functions. Differential operators can be consistently approximated on the evolving, irregular set of particles using the DC-PSE scheme [39], which can be seen as a generalization of vorticity redistribution schemes [41,4,25] to arbitrary differential operators.…”

“…The operator kernels η p (not to be confused with the basis functions ζ p (x; x p ) above) are discretized versions of the generalized integral operators proposed by Eldredge et al [17] and depend on the desired order of accuracy (see Schrader et al [39] and Appendix A).…”