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

Abstract

Abstract. We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in 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 computational cost that can be achieved with this method over the standard (explicit) RBF-FD method are demonstrated with a series of numerical examples. We also illustrate the flexibility and general applicability of the method by solving two different reaction-diffusion equations on surfaces that are defined implicitly and only by point clouds.

RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for solving PDEs on surfaces

“…Unfortunately, unlike in [25,38], we were unable to find stable parameters for point cloud models of more complicated manifolds such as frogs and bunnies. It is likely that such surfaces would require an adaptive tolerance selection for the LOI procedure, which we leave for future work.…”

“…To avoid differentiating normals, we will accomplish this using iterated interpolation [24,25,38]. This is done in two steps: first compute overlapped RBF-FD weights for the operators G x ,G y , and G z at all stencil points x I 1 j (every point in P 1 ); then, combine those RBF-FD weights in such a way that we only compute the weights for all nodes with indices in the set R 1 .…”

“…In [38], this stability was achieved by performing a nonlinear optimization for the shape parameter on each stencil, constrained so that the RBF interpolation matrices on each stencil should have approximately the same condition number. In [25], this stability was achieved by encouraging diagonal dominance in RBF-HFD differentiation matrices using a stencil selection algorithm.…”

“…RBF-based methods also generalize naturally to the solution of PDEs on manifolds M ⊂ R 3 using only the Euclidean distance measure in the embedding space and Cartesian coordinates. This ability has been leveraged to obtain four important classes of methods for solving PDEs on manifolds: global RBF methods [17,18,24,29], RBF-Finite Difference (RBF-FD) methods [16,20,30,38], RBF-Partition of Unity (RBF-PU) methods [1], and implicit/Hermite RBF-FD methods [25]. We will focus on RBF-FD methods for M ⊂ R 3 for the remainder of this article.…”

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

“…RBF interpolants can be used to generate both pseudospectral (RBF-PS) and finite-difference (RBF-FD) methods [1][2][3][4][5][6]. RBF-based methods are also easily applied to the solution of PDEs on node sets that are not unisolvent for polynomials, such as ones lying on the sphere S 2 [7][8][9][10] and other general surfaces [11][12][13][14][15].…”