Fourth order partial differential equations on general geometries
Journal of Computational Physics
Maintaining the point correspondence in the level set framework
Journal of Computational Physics
Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology
Journal of Computational Physics
A simple embedding method for solving partial differential equations on surfaces
Journal of Computational Physics
Level Set Equations on Surfaces via the Closest Point Method
Journal of Scientific Computing
Journal of Scientific Computing
A Finite Volume Method for Solving Parabolic Equations on Logically Cartesian Curved Surface Meshes
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Journal of Computational Physics
A Convergent Finite Volume Scheme for Diffusion on Evolving Surfaces
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Real-Time Fluid Effects on Surfaces using the Closest Point Method
Computer Graphics Forum
Method of Moving Frames to Solve Conservation Laws on Curved Surfaces
Journal of Scientific Computing
Closest point turbulence for liquid surfaces
ACM Transactions on Graphics (TOG)
Journal of Computational Physics
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We improve upon a method introduced in Bertalmio et al. [4] for solving evolution PDEs on codimension-one surfaces in $$\mathbb{R}^N.$$ As in the original method, by representing the surface as a level set of a smooth function, we use only finite differences on a Cartesian mesh to solve an Eulerian representation of the surface PDE in a neighborhood of the surface. We modify the original method by changing the Eulerian representation to include effects due to surface curvature. This modified PDE has the very useful property that any solution which is initially constant perpendicular to the surface remains so at later times. 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. We present numerical examples that include convergence tests in neighborhoods of the surface that shrink with the grid size