An improved geometry-aware curvature discretization for level set methods: Application to tumor growth

An advantage of using level set methods for moving boundary problems is that geometric quantities such as curvature can be readily calculated from the level set function. However, in topologically challenging cases (e.g., when two interfaces are in close contact), level set functions develop singularities that yield inaccurate curvatures when using traditional discretizations. In this note, we give an improved discretization of curvature for use near level set singularities. Where level set irregularities are detected, we use a local polynomial approximation of the interface to construct the level set function on a local subgrid, where we can accurately calculate the curvature using the standard 9-point discretization. We demonstrate that this new algorithm is capable of calculating the curvature accurately in a variety of situations where the traditional algorithm fails and provide numerical evidence that the method is second-order accurate. Examples are drawn from modified Hele-Shaw flows and a model of solid tumor growth.