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

@article{Fuselier2013AHK, title={A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces}, author={Edward J. Fuselier and Grady B. Wright}, journal={Journal of Scientific Computing}, year={2013}, volume={56}, pages={535-565} }

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 $$\mathbb{R }^d$$. For two-dimensional surfaces embedded in $$\mathbb{R }^3$$, 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…

## Figures, Tables, and Topics from this paper

## 86 Citations

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

- Mathematics, MedicineJ. Sci. Comput.
- 2015

This paper presents 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 method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation.

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

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2017

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 is presented.

Approximation of continuous surface differential operators with the generalized moving least-squares (GMLS) method for solving reaction–diffusion equation

- MathematicsComputational and Applied Mathematics
- 2018

In this paper, a meshless approximation based on generalized moving least squares is applied to solve the reaction–diffusion equations on the sphere and red-blood cell surfaces. The proposed method…

Kernel-based methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds

- Computer Science, MathematicsArXiv
- 2021

This paper establishes the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type, and provides numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold.

A lifted local Galerkin method for solving the reaction-diffusion equations on implicit surfaces

- Computer Science, MathematicsComput. Phys. Commun.
- 2018

A lifted local Galerkin method for solving the reaction–diffusion equations on implicit surfaces by discretizing the equations on tangent plane of each surface node by a local Galerskin method is presented.

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

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2018

A finite element method with mass lumping (LESFEM) and the existence of a family of rectangles in the phase space that are invariant only under specific growth laws for the well-known activator-depleted and Thomas reaction–diffusion models is proved.

A local radial basis function method for the Laplace-Beltrami operator

- Computer Science, MathematicsJ. Sci. Comput.
- 2021

A new local meshfree method for the approximation of the Laplace-Beltrami operator on a smooth surface of co-dimension one embedded in $\R^3$.

Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD

- Mathematics
- 2021

The Gray–Scott (GS) model is a non-linear system of equations generally adopted to describe reaction–diffusion dynamics. In this paper, we discuss a numerical scheme for solving the GS system. The…

Solving partial differential equations on (evolving) surfaces with radial basis functions

- Computer Science, MathematicsAdv. Comput. Math.
- 2020

Three different meshfree kernel-based methods for the spatial discretisation of semi-linear parabolic partial differential equations on surfaces, i.e. on smooth, compact, connected, orientable, and closed (d − 1)-dimensional submanifolds of $\mathbb {R}^{d}$ .

Generalized finite difference method for anomalous diffusion on surfaces

- Physics
- 2021

In this study, a localized collocation method called generalized finite difference method (GFDM) is developed to solve the anomalous diffusion problems on surfaces. The expressions of the surface…

## References

SHOWING 1-10 OF 85 REFERENCES

The Implicit Closest Point Method for the Numerical Solution of Partial Differential Equations on Surfaces

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2009

A new implicit closest point method for surface PDEs that allows for large, stable time steps while retaining the principal benefits of the original method and works on sharply defined bands without degrading the accuracy of the method.

Level Set Equations on Surfaces via the Closest Point Method

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2008

The main modification is to introduce a Weighted Essentially Non-Oscillatory (WENO) interpolation step into the Closest Point Method, which gives high-order results on a variety of smooth test problems including passive transport, normal flow and redistancing.

A simple embedding method for solving partial differential equations on surfaces

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2008

A simple method for the numerical solution of partial differential equations which embeds the problem within a Cartesian analog of the original equation, posed on the entire space containing the surface.

An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2006

The change remedies many of problems facing the original method, including a need to frequently extend data off of the surface, uncertain boundary conditions, and terribly degenerate parabolic PDEs.

A Finite Volume Method for Solving Parabolic Equations on Logically Cartesian Curved Surface Meshes

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2009

A second order finite volume scheme for the constant-coefficient diffusion equation on curved parametric surfaces that can be easily coupled to existing finite volume solvers for logically Cartesian meshes, and handles general mixed boundary conditions is presented.

Finite elements on evolving surfaces

- Mathematics
- 2007

In this article, we define a new evolving surface finite-element method for numerically approximating partial differential equations on hypersurfaces (t) in n+1 which evolve with time. The key idea…

A geometrical approach to wave-type solutions of excitable reaction-diffusion systems

- MathematicsProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
- 1991

We formulate the eikonal equation approximation for travelling waves in excitable reaction-diffusion systems, which have been proposed as models for a large number of biomedical situations. This…

Implicit-explicit methods for time-dependent partial differential equations

- Mathematics
- 1995

Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized partial differential equations (PDEs) of…

A multigrid finite element method for reaction-diffusion systems on surfaces

- Mathematics, Computer ScienceComput. Vis. Sci.
- 2010

A multigrid finite element approach to solve PDE’s on surfaces that allows to reuse code initially developed to solve problems on planar domains to solve the corresponding problem on surfaces.

Surface Finite Elements for Parabolic Equations

- 2007

In this article we define a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces Γ in R. The key idea is based on the…