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

DOI10.1137/16M1095457zbMath1371.41018OpenAlexW2756899824MaRDI QIDQ5358964

Erik Lehto, Grady B. Wright, Varun Shankar

Publication date: 22 September 2017

Published in: SIAM Journal on Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/16m1095457

zbMATH Keywords

manifoldsRBF-FDreaction diffusionRBF-HFD

Mathematics Subject Classification ID

Ill-posedness and regularization problems in numerical linear algebra (65F22) Padé approximation (41A21) Numerical interpolation (65D05) General theory of numerical methods in complex analysis (potential theory, etc.) (65E05)

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Uses Software

• Matlab
• MeshLab
• DistMesh
• rbf_qr

Cites Work

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• A high-order kernel method for diffusion and reaction-diffusion equations on surfaces
• The orthogonal gradients method: a radial basis functions method for solving partial differential equations on arbitrary surfaces
• A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere
• Stable calculation of Gaussian-based RBF-FD stencils
• Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions
• RBF-FD formulas and convergence properties
• Adaptive meshless centres and RBF stencils for Poisson equation
• Stabilization of RBF-generated finite difference methods for convective PDEs
• On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs
• On the role of polynomials in RBF-FD approximations. I: Interpolation and accuracy
• The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems
• Compact finite difference schemes with spectral-like resolution
• Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier--Stokes equations
• Interpolation in the limit of increasingly flat radial basis functions
• Stable computation of multiquadric interpolants for all values of the shape parameter
• Approximation of parabolic PDEs on spheres using spherical basis functions
• Order-preserving derivative approximation with periodic radial basis functions
• A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction-diffusion equations on surfaces
• Multivariate interpolation by polynomials and radial basis functions
• The Runge phenomenon and spatially variable shape parameters in RBF interpolation
• Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions
• Scattered node compact finite difference-type formulas generated from radial basis functions
• Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions
• Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
• Stable Evaluation of Gaussian Radial Basis Function Interpolants
• The Implicit Closest Point Method for the Numerical Solution of Partial Differential Equations on Surfaces
• Stable Computations with Gaussian Radial Basis Functions
• A Primer on Radial Basis Functions with Applications to the Geosciences
• A radial basis function method for the shallow water equations on a sphere
• Local RBF-FD solutions for steady convection–diffusion problems
• A Stable Algorithm for Flat Radial Basis Functions on a Sphere
• Gershgorin Theory for the Generalized Eigenvalue Problem Ax = λBx
• Generalized Hermite Interpolation Via Matrix-Valued Conditionally Positive Definite Functions
• A Simple Mesh Generator in MATLAB
• Implicit-Explicit Methods for Time-Dependent Partial Differential Equations
• Localization of generalized eigenvalues by Cartesian ovals

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