| We present a novel class of particle methods with deformable shapes that achieve high-order convergence rates in the supremum norm. These methods do not require remappings or extended overlapping or vanishing moments for the particles. Unlike classical convergence analysis, our estimates do not rely on a smoothing kernel argument 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 approximations" 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 error estimates that demonstrate its uniform convergence and the uniform boundedness of the particle overlapping. To illustrate the flexibility of the method we also develop an adaptive multilevel version that includes a local correction filter for positivity-preserving hierarchical approximations. Numerical studies demonstrate the convergence properties of this new particle scheme in both its uniform and adaptive versions, and compare it with traditional fixed-shape particle methods with or without remappings. |
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