Grid-free adaptive semi-Lagrangian advection using radial basis functions

This paper proposes a new grid-free adaptive advection scheme. The resulting algorithm is a combination of the semi-Lagrangian method (SLM) and the grid-free radial basis function interpolation (RBF). The set of scattered interpolation nodes is subject to dynamic changes at run time. Based on a posteriori local error estimates, a self-adaptive local refinement and coarsening of the nodes serves to obtain enhanced accuracy at reasonable computational costs. Due to well-known features of SLM and RBF, the method is guaranteed to be stable, it has good approximation behaviour, and it works for arbitrary space dimension. Numerical examples in two dimensions illustrate the performance of the method in comparison with existing grid-based advection schemes.