Shinnosuke Obi

Matthias Kirchhart, Shinnosuke Obi

We present a splitting-free variant of the vorticity redistribution method. Spatial consistency and stability when combined with a time-stepping scheme are proven. We propose a new strategy preventing excessive growth in the number of particles while retaining the order of consistency. The novel concept of small neighbourhoods significantly reduces the method's computational cost. In numerical experiments the method showed second order convergence, one order higher than predicted by the analysis. Compared to the fast multipole code used in the velocity computation, the method is about three times faster.

Matthias Kirchhart, Shinnosuke Obi

We present a new class of $C^\infty$-smooth finite element spaces on Cartesian grids, based on a partition of unity approach. We use these spaces to construct smooth approximations of particle fields, i.e., finite sums of weighted Dirac deltas. In order to use the spaces on general domains, we propose a fictitious domain formulation, together with a new high-order accurate stabilization. Stability, convergence, and conservation properties of the scheme are established. Numerical experiments confirm the analysis and show that the Cartesian grid-size $σ$ should be taken proportional to the square-root of the particle spacing $h$, resulting in significant speed-ups in vortex methods.