Smooth particle methods without smoothing

Details Smooth particle methods without smoothing Smooth particle methods without smoothing

2011-12-08

arxiv.org/abs/1112.1859v2

Mathematics

Numerical Analysis

second version

Scientific paper

We present a novel class of particle methods with deformable shapes that achieve high-order convergence rates in the uniform norm without requiring remappings, extended overlapping or vanishing moments for the particles. Unlike classical convergence analysis, our estimates do not rely on the use of a smoothing kernel but rather on the uniformly bounded overlapping of the particles supports and on the smoothness of the characteristic flow. In particular, they also apply to heterogeneous particle decompositions such as piecewise polynomial bases on unstructured meshes. In the first-order case which simply consists of pushing forward linearly transformed particles (LTP) along the flow, we provide an explicit scheme and establish rigorous estimates that demonstrate the convergence in the uniform norm and the uniform boundedness of the resulting particle overlapping. To illustrate the flexibility of the method we develop an adaptive multilevel version where particles are dynamically refined based on a local error indicator. Numerical studies allow to assess the convergence properties of this new particle scheme in both its uniform and adaptive versions, by comparing it with traditional fixed-shape particle methods with or without remappings.

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