Adaptive radial basis function generated finite-difference (RBF-FD) on non-uniform nodes using pp-refinement

Radial basis functions-generated finite difference methods (RBF-FDs) have been gaining popularity recently. In particular, the RBF-FD based on polyharmonic splines (PHS) augmented with multivariate polynomials (PHS+poly) has been found significantly effective. For the approximation order of RBF-FDs' weights on scattered nodes, one can already find mathematical theories in the literature. Many practical problems in numerical analysis, however, do not require a uniform node-distribution. Instead, they would be better suited if specific areas of the domain, where complicated physics needed to be resolved, had a relatively higher node-density compared to the rest of the domain. In this work, we proposed a practical adaptive RBF-FD with a user-defined order of convergence with respect to the total number of (possibly scattered and non-uniform) data points $N$. Our algorithm outputs a sparse differentiation matrix with the desired approximation order. Numerical examples are provided to show that the proposed adaptive RBF-FD method yields the expected $N$-convergence even for highly non-uniform node-distributions. The proposed method also reduces the number of non-zero elements in the linear system without sacrificing accuracy.Comment: An updated version with seismic modeling will be included in version

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

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