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

Abstract

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.

Conservative semi‐Lagrangian advection on adaptive unstructured meshes

“…The following construction of our method combines the particle-based semi-Lagrangian approach of the previous papers [1,2,3] with a finite volume scheme on adaptive unstructured meshes. We remark that the particlebased scheme of [1] is applied in [2] on a real-world test case scenario concerning chemical tracer transportation over the Arctic stratosphere.…”

“…We remark that the particlebased scheme of [1] is applied in [2] on a real-world test case scenario concerning chemical tracer transportation over the Arctic stratosphere. An extension of the scheme in [1] to nonlinear transport equations has recently been suggested in [3].…”

“…We remark that the particlebased scheme of [1] is applied in [2] on a real-world test case scenario concerning chemical tracer transportation over the Arctic stratosphere. An extension of the scheme in [1] to nonlinear transport equations has recently been suggested in [3]. However, none of the meshfree particle schemes in [1,2,3] is conservative, which is a severe drawback in several relevant applications, such as the above mentioned tracer advection simulation.…”

“…Similar to [1,2,3], the discretization works with a finite set Ξ ⊂ Ω of nodes, each of which corresponds to one flow particle at a time t ∈ I. Any current node set Ξ defines a unique Voronoi diagram V Ξ , which yields a partitioning of the computational domain Ω = R 2 into #Ξ finite control volumes, Voronoi cells, so that each cell in V Ξ contains exactly one node from Ξ.…”

“…We have combined the conservative advection scheme of this paper with the ideas of the adaption strategy in [1,2,3], in order to obtain an adaptive and conservative semi-Lagrangian advection method. In order to keep this paper widely self-contained, we wish to discuss the basic ideas of the node adaption strategy of the previous work [2] in this section.…”