Scattered Data Approximation

Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.

Solving PDEs with radial basis functions

On the role of polynomials in RBF-FD approximations

Non-intrusive reduced-order modelling of the Navier-Stokes equations based on RBF interpolation

In this paper, we present a method based on radial basis function (RBF)-generated finite differences (FD) for numerically solving diffusion and reaction---diffusion equations (PDEs) on closed surfaces embedded in $${\mathbb {R}}^d$$Rd. Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.

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A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction---Diffusion Equations on Surfaces

We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in $\mathbb{R}^d$. The novelty of the method is in the approximation of the Laplace--Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace--Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computa...

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A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion equations

The overlapped radial basis function-finite difference (RBF-FD) method: A generalization of RBF-FD

We present a new algorithm for the automatic one-shot generation of scattered node sets on irregular two dimensional (2D) and three dimensional (3D) domains using Poisson disk sampling coupled to n...

Varun Shankar

Papers

A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction---Diffusion Equations on Surfaces

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion equations

The overlapped radial basis function-finite difference (RBF-FD) method: A generalization of RBF-FD

Robust Node Generation for Mesh-free Discretizations on Irregular Domains and Surfaces