Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
A fast level set method for propagating interfaces
Journal of Computational Physics
A PDE-based fast local level set method
Journal of Computational Physics
Numerical Methods
Numerical approximations of singular source terms in differential equations
Journal of Computational Physics
Discretization of Dirac delta functions in level set methods
Journal of Computational Physics
Two methods for discretizing a delta function supported on a level set
Journal of Computational Physics
Geometric integration over irregular domains with application to level-set methods
Journal of Computational Physics
A convergence rate theorem for finite difference approximations to delta functions
Journal of Computational Physics
Journal of Computational Physics
Discretizing delta functions via finite differences and gradient normalization
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.45 |
We present a finite difference method for discretizing a Heaviside function H(u(x-)), where u is a level set function u:R^n@?R that is positive on a bounded region @W@?R^n. There are two variants of our algorithm, both of which are adapted from finite difference methods that we proposed for discretizing delta functions in [J.D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007) 915-931; J.D. Towers, Discretizing delta functions via finite differences and gradient normalization, Preprint at http://www.miracosta.edu/home/jtowers/; J.D. Towers, A convergence rate theorem for finite difference approximations to delta functions, J. Comput. Phys. 227 (2008) 6591-6597]. We consider our approximate Heaviside functions as they are used to approximate integrals over @W. We prove that our first approximate Heaviside function leads to second order accurate quadrature algorithms. Numerical experiments verify this second order accuracy. For our second algorithm, numerical experiments indicate at least third order accuracy if the integrand f and @?@W are sufficiently smooth. Numerical experiments also indicate that our approximations are effective when used to discretize certain singular source terms in partial differential equations. We mostly focus on smooth f and u. By this we mean that f is smooth in a neighborhood of @W, u is smooth in a neighborhood of @?@W, and the level set u(x)=0 is a manifold of codimension one. However, our algorithms still give reasonable results if either f or u has jumps in its derivatives. Numerical experiments indicate approximately second order accuracy for both algorithms if the regularity of the data is reduced in this way, assuming that the level set u(x)=0 is a manifold. Numerical experiments indicate that dependence on the placement of @W with respect to the grid is quite small for our algorithms. Specifically, a grid shift results in an O(h^p) change in the computed solution, where p is the observed rate of convergence.