An Improvement of a Recent Eulerian Method for Solving PDEs on General Geometries

We improve upon a method introduced in (Bertalmio et. al. JCP 2001) for solving evolution PDEs on codimension-one surfaces in RN . 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.