Generating textures on arbitrary surfaces using reaction-diffusion
Proceedings of the 18th annual conference on Computer graphics and interactive techniques
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
The visualization toolkit (2nd ed.): an object-oriented approach to 3D graphics
The visualization toolkit (2nd ed.): an object-oriented approach to 3D graphics
Variational problems and partial differential equations on implicit surfaces
Journal of Computational Physics
Numerical Initial Value Problems in Ordinary Differential Equations
Numerical Initial Value Problems in Ordinary Differential Equations
Motion of curves constrained on surfaces using a level-set approach
Journal of Computational Physics
Flows on surfaces of arbitrary topology
ACM SIGGRAPH 2003 Papers
Semi-Implicit Level Set Methods for Curvature and Surface Diffusion Motion
Journal of Scientific Computing
Fourth order partial differential equations on general geometries
Journal of Computational Physics
An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries
Journal of Scientific Computing
Python for Scientific Computing
Computing in Science and Engineering
A simple embedding method for solving partial differential equations on surfaces
Journal of Computational Physics
A Local Semi-Implicit Level-Set Method for Interface Motion
Journal of Scientific Computing
Level Set Equations on Surfaces via the Closest Point Method
Journal of Scientific Computing
Optimal implicit strong stability preserving Runge--Kutta methods
Applied Numerical Mathematics
Finite element approximation of elliptic partial differential equations on implicit surfaces
Computing and Visualization in Science
Finite Element Methods on Very Large, Dynamic Tubular Grid Encoded Implicit Surfaces
SIAM Journal on Scientific Computing
Segmentation on surfaces with the closest point method
ICIP'09 Proceedings of the 16th IEEE international conference on Image processing
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Journal of Computational Physics
A level set projection model of lipid vesicles in general flows
Journal of Computational Physics
CAD and mesh repair with Radial Basis Functions
Journal of Computational Physics
Real-Time Fluid Effects on Surfaces using the Closest Point Method
Computer Graphics Forum
Closest point turbulence for liquid surfaces
ACM Transactions on Graphics (TOG)
Journal of Scientific Computing
Journal of Computational Physics
The Explicit-Implicit-Null method: Removing the numerical instability of PDEs
Journal of Computational Physics
A level-set method for two-phase flows with moving contact line and insoluble surfactant
Journal of Computational Physics
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Many applications in the natural and applied sciences require the solutions of partial differential equations (PDEs) on surfaces or more general manifolds. The closest point method is a simple and accurate embedding method for numerically approximating PDEs on rather general smooth surfaces. However, the original formulation is designed to use explicit time stepping. This may lead to a strict time-step restriction for some important PDEs such as those involving the Laplace-Beltrami operator or higher-order derivative operators. To achieve improved stability and efficiency, we introduce a new implicit closest point method for surface PDEs. The method allows for large, stable time steps while retaining the principal benefits of the original method. In particular, it maintains the order of accuracy of the discretization of the underlying embedding PDE, it works on sharply defined bands without degrading the accuracy of the method, and it applies to general smooth surfaces. It also is very simple and may be applied to a rather general class of surface PDEs. Convergence studies for the in-surface heat equation and a fourth-order biharmonic problem are given to illustrate the accuracy of the method. We demonstrate the flexibility and generality of the method by treating flows involving diffusion, reaction-diffusion, and fourth-order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension.