Discretization-Corrected PSE Operators For Adaptive Multiresolution Particle Methods

Mathematical models in the form of differential equations can often not be solved analytically, but need to be simulated by means of a numerical method. Particle methods are mesh-free schemes that use independent or pairwise interacting particles to represent the physical properties of a system. This allows for natural adaptivity in complex or deforming geometries. The Lagrangian frame of reference when tracking the particles during a simulation renders particle methods particularly successful in the area of fluid mechanics. Their numerical stability in advection-dominated problems is superior to that of descriptions in an Eulerian frame of reference. The general particle strength exchange (PSE) operators [31] approximate derivatives on scattered particle locations to any desired order of accuracy. Convergence, however, is limited by the discretization error resulting from an inherent numerical quadrature. In this thesis, we introduce a consistent discretization correction framework for PSE operators. With this correction, the operators yield the desired rate of convergence for any resolution, both on uniform Cartesian and irregular particle distributions, as well as near boundaries. The discretization correction allows setting the kernel width to arbitrarily small values for constant interparticle spacing. We show that, on uniform Cartesian particle distributions, this leads to a seamless transition from discretizationcorrected (DC) PSE operators to classical finite-difference stencils. We further identify relationships between DC PSE operators and operators used in corrected smoothed particle hydrodynamics, reproducing kernel