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 timeintegration, 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.

A mesh‐free radial basis function–based semi‐Lagrangian lattice Boltzmann method for incompressible flows

“…20 Moreover, it is easy to find the departure point in the LBM, ie, position x − e t. When the nonuniform grid is used, the departure point may not be the grid point in general. 20 Moreover, it is easy to find the departure point in the LBM, ie, position x − e t. When the nonuniform grid is used, the departure point may not be the grid point in general.…”

“…Because the advection operator (ie, the streaming process) of the LBM is linear, 1 the prestreaming distribution function at an Eulerian point equals to that at its departure point. 20 Moreover, it is easy to find the departure point in the LBM, ie, position x − e t. When the nonuniform grid is used, the departure point may not be the grid point in general. Thus, the distribution function at the departure point can be achieved by interpolating the distribution functions at some grid points.…”

“…In the local RBF interpolation, only the n-1 nearest neighbors of knot x i and itself are needed and n ≪ N. 20 Moreover, in order to improve the accuracy of RBF interpolation, the polynomial terms 28 Thus, Equation (10) becomes In the local RBF interpolation, only the n-1 nearest neighbors of knot x i and itself are needed and n ≪ N. 20 Moreover, in order to improve the accuracy of RBF interpolation, the polynomial terms 28 Thus, Equation (10) becomes…”

“…Considering the computational expenses, the local RBF interpolation 20 is used in this paper. In the local RBF interpolation, only the n-1 nearest neighbors of knot x i and itself are needed and n ≪ N. 20 Moreover, in order to improve the accuracy of RBF interpolation, the polynomial terms 28 Thus, Equation (10) becomes…”

“…However, the streaming step cannot be carried out directly because the Eulerian points do not coincide with the positions where the distribution functions are streamed. Inspired by the work reported in the literature, 19,20 the streaming step can be solved in a semi-Lagrangian framework. In order to obtain the prestreaming distribution functions at the Eulerian points, a series of Lagrangian points should be constructed firstly, which are used as the departure points of the streaming step.…”