Mesh-free Semi-Lagrangian Methods for Transport on a Sphere Using Radial Basis Functions

TL;DR

This paper introduces three mesh-free semi-Lagrangian methods using radial basis functions for simulating transport on a sphere, avoiding coordinate singularities and stabilization issues, with demonstrated high accuracy and efficiency.

Contribution

The paper develops three novel mesh-free semi-Lagrangian RBF-based methods for spherical transport, incorporating spherical harmonics and comparing their performance.

Findings

Global RBFs achieve spectral convergence for smooth solutions.

Local RBF stencil and partition of unity methods attain high-order accuracy.

Methods effectively handle transport problems with reduced parameter tuning.

Abstract

We present three new semi-Lagrangian methods based on radial basis function (RBF) interpolation for numerically simulating transport on a sphere. The methods are mesh-free and are formulated entirely in Cartesian coordinates, thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface-based coordinate systems. For problems involving tracer transport in a given velocity field, the semi-Lagrangian framework allows these new methods to avoid the use of any stabilization terms (such as hyperviscosity) during time-integration, thus reducing the number of parameters that have to be tuned. The three new methods are based on interpolation using 1) global RBFs, 2) local RBF stencils, and 3) RBF partition of unity. For the latter two of these methods, we find that it is crucial to include some low degree spherical harmonics in the interpolants. Standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy, efficiency, conservation properties, and dissipation/dispersion errors. For global RBFs, spectral spatial convergence is observed for smooth solutions on quasi-uniform nodes, while high-order accuracy is observed for the local RBF stencil and partition of unity approaches.