Convergence of finite volume schemes for Poisson's equation on nonuniform meshes
SIAM Journal on Numerical Analysis
Generating textures on arbitrary surfaces using reaction-diffusion
Proceedings of the 18th annual conference on Computer graphics and interactive techniques
RKC: an explicit solver for parabolic PDEs
Journal of Computational and Applied Mathematics
Implicit fairing of irregular meshes using diffusion and curvature flow
Proceedings of the 26th annual conference on Computer graphics and interactive techniques
A finite volume method for the approximation of diffusion operators on distorted meshes
Journal of Computational Physics
Analysis of the cell-centred finite volume method for the diffusion equation
Journal of Computational Physics
Variational problems and partial differential equations on implicit surfaces
Journal of Computational Physics
Variational mesh adaptation: isotropy and equidistribution
Journal of Computational Physics
Transport and diffusion of material quantities on propagating interfaces via level set methods
Journal of Computational Physics
An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries
Journal of Scientific Computing
Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology
Journal of Computational Physics
Finite-volume transport on various cubed-sphere grids
Journal of Computational Physics
Journal of Computational Physics
A simple embedding method for solving partial differential equations on surfaces
Journal of Computational Physics
An Adaptive Finite Element Method for the Laplace-Beltrami Operator on Implicitly Defined Surfaces
SIAM Journal on Numerical Analysis
A discrete scheme of Laplace-Beltrami operator and its convergence over quadrilateral meshes
Computers & Mathematics with Applications
Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Classics in Applied Mathematics Classics in Applied Mathemat)
A Nine Point Scheme for the Approximation of Diffusion Operators on Distorted Quadrilateral Meshes
SIAM Journal on Scientific Computing
Finite Volume Method for 2D Linear and Nonlinear Elliptic Problems with Discontinuities
SIAM Journal on Numerical Analysis
Discrete Laplace--Beltrami operators and their convergence
Computer Aided Geometric Design
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We present a second order finite volume scheme for the constant-coefficient diffusion equation on curved parametric surfaces. While our scheme is applicable to general quadrilateral surface meshes based on smooth or piecewise smooth coordinate transformations, our primary motivation for developing the present scheme is to solve diffusion problems on a particular set of circular and spherical meshes introduced in [D. A. Calhoun, C. Helzel, and R. J. LeVeque, SIAM Rev., 50 (2008), pp. 723-752] for the discretization of hyperbolic problems. These grids are generated from mappings of a single Cartesian grid and were designed to have nearly uniform cells sizes and avoid the pole singularity associated with polar or spherical grid mappings. The present method for parabolic equations offers several advantages. It does not require analytic metric terms, shows second order accuracy on our disk and sphere grids, can be easily coupled to existing finite volume solvers for logically Cartesian meshes, and handles general mixed boundary conditions. Our parabolic scheme should appeal to researchers in the fields of geophysical fluid dynamics, computational biology, and any other discipline that requires the solution of parabolic equations on quadrilateral surface meshes. In this article, we present several numerical examples demonstrating the accuracy of the scheme, and then use the scheme to solve advection-reaction-diffusion equations modeling biological pattern formation on surfaces.