A level set approach for computing solutions to incompressible two-phase flow
Journal of Computational Physics
A fast level set method for propagating interfaces
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
A variational level set approach to multiphase motion
Journal of Computational Physics
Fitting smooth surfaces to dense polygon meshes
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Automatic reconstruction of B-spline surfaces of arbitrary topological type
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
SIAM Journal on Scientific Computing
Thin Films with High Surface Tension
SIAM Review
Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility
Mathematics of Computation
On an Alternating Direction Method for Solving the Plate Problem with Mixed Boundary Conditions
Journal of the ACM (JACM)
A PDE-based fast local level set method
Journal of Computational Physics
Positivity-Preserving Numerical Schemes for Lubrication-Type Equations
SIAM Journal on Numerical Analysis
Adaptively sampled distance fields: a general representation of shape for computer graphics
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Variational problems and partial differential equations on implicit surfaces
Journal of Computational Physics
Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces: 730
Journal of Computational Physics
Anisotropic diffusion of surfaces and functions on surfaces
ACM Transactions on Graphics (TOG)
Transport and diffusion of material quantities on propagating interfaces via level set methods
Journal of Computational Physics
ADI schemes for higher-order nonlinear diffusion equations
Applied Numerical Mathematics
Semi-Implicit Level Set Methods for Curvature and Surface Diffusion Motion
Journal of Scientific Computing
An Eulerian Formulation for Solving Partial Differential Equations Along a Moving Interface
Journal of Scientific Computing
Journal of Computational Physics
Energy-minimizing splines in manifolds
ACM SIGGRAPH 2004 Papers
Discretization of Dirac delta functions in level set methods
Journal of Computational Physics
An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries
Journal of Scientific Computing
Processing textured surfaces via anisotropic geometric diffusion
IEEE Transactions on Image Processing
Finite elements on point based surfaces
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
A simple embedding method for solving partial differential equations on surfaces
Journal of Computational Physics
On the numerical solution of a driven thin film equation
Journal of Computational Physics
Journal of Scientific Computing
SIAM Journal on Scientific Computing
A Prolate-Element Method for Nonlinear PDEs on the Sphere
Journal of Scientific Computing
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Journal of Computational Physics
Higher order numerical discretizations for exterior and biharmonic type PDEs
Journal of Computational and Applied Mathematics
Image Denoising Using Mean Curvature of Image Surface
SIAM Journal on Imaging Sciences
Journal of Computational Physics
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We extend a recently introduced method for numerically solving partial differential equations on implicit surfaces [M. Bertalmío, L.T. Cheng, S. Osher, G. Sapiro. Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys. 174 (2) (2001) 759-780] to fourth order PDEs including the Cahn-Hilliard equation and a lubrication model for curved surfaces. By representing a surface in RN as the level set of a smooth function, φ, we compute the PDE using only finite differences on a standard Cartesian mesh in RN. The higher order equations introduce a number of challenges that are of less concern when applying this method to first and second order PDEs. Many of these problems, such as time-stepping restrictions and large stencil sizes, are shared by standard fourth order equations in Euclidean domains, but others are caused by the extreme degeneracy of the PDEs that result from this method and the general geometry. We approach these difficulties by applying convexity splitting methods, ADI schemes, and iterative solvers. We discuss in detail the differences between computing these fourth order equations and computing the first and second order PDEs considered in earlier work. We explicitly derive schemes for the linear fourth order diffusion, the Cahn-Hilliard equation for phase transition in a binary alloy, and surface tension driven flows on complex geometries. Numerical examples validating our methods are presented for these flows for data on general surfaces.