A Particle Method without Remeshing

We propose a simple tweak to a recently developed regularisation scheme for particle methods. This allows us to chose the particle spacing $h$ proportional to the regularisation length $σ$ and achieve optimal error bounds of the form $\mathcal{O}(σ^n)$, $n\in\mathbb{N}$, without any need of remeshing. We prove this result for the linear advection equation but also carry out high-order experiments on the full Navier--Stokes equations. In our experiments the particle methods proved to be highly accurate, long-term stable, and competitive with discontinuous Galerkin methods.

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