A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

Abstract

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in R d . For two-dimensional surfaces embedded in R 3 , these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.

Numerical Preservation of Velocity Induced Invariant Regions for Reaction–Diffusion Systems on Evolving Surfaces

“…Now, we consider two different RDSs that are well-known in the literature and prove, at the discrete level, the existence of discrete invariant hyper-rectangles for these RDSs, depending on the global discrete dilation rates μ min and μ max defined in (18). The results in this section are confined to the spatially discrete level, but from Conjecture 1, we claim that the same results holds at the continuous level.…”

“…where μ * min and μ * max are defined in (18). Next, we conjecture a criterion under which a hyper-rectangle is invariant for system (16).…”

“…Among the applications of RDSs on surfaces we mention brain growth [26], cell migration [3], chemotaxis [12], developmental biology [28], electrodeposition [24] and phase field modeling [42]. The growing interest toward PDEs on evolving surfaces has stimulated the development of several numerical methods for such problems, among which we mention (but not limited to) embedding methods [2], kernel methods [18], implicit boundary integral methods [5,35], surface finite element methods (SFEM) [10] and some of their recent variations and extensions [13,16,17,20,23,40].…”

A radial basis function (RBF) finite difference method for the simulation of reaction–diffusion equations on stationary platelets within the augmented forcing method

“…The approach we use to approximate the surface Laplacian mimics the formulation given in (12) and is conceptually similar to that based on global RBFs given in [26]. It is worth noting at this point that though the normal vector is obtained from the parametric representation of the platelet, one could certainly use normal vectors derived from level set representations or, more generally, signeddistance representations of the data sites.…”

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

“…Global radial basis function (RBF) methods are quite popular for the numerical solution of various partial differential equations (PDEs) due to their ability to handle scattered node layouts, their simplicity of implementation, and their potential for spectral accuracy for smooth problems. These methods have been successfully applied to the solution of PDEs in various geometries in R 2 and R 3 (e.g., [12,17]), including spherical domains (e.g., [27,16,42]), and more general surfaces embedded in R 3 (e.g., [26,36]). When high orders of algebraic accuracy are sufficient for a given problem, or if the solutions to the problem are expected to only have finite smoothness, RBF generated finite difference (RBF-FD) formulas are an attractive alternative to global RBFs as they perform better in terms of accuracy per computational cost [17].…”