Title: Mesh-free Semi-Lagrangian Methods for Transport on a Sphere Using Radial Basis Functions,
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.