Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Analysis of a one-dimensional model for the immersed boundary method
SIAM Journal on Numerical Analysis
A PDE-based fast local level set method
Journal of Computational Physics
Regularization Techniques for Numerical Approximation of PDEs with Singularities
Journal of Scientific Computing
Numerical approximations of singular source terms in differential equations
Journal of Computational Physics
The numerical approximation of a delta function with application to level set methods
Journal of Computational Physics
High order numerical methods to a type of delta function integrals
Journal of Computational Physics
Short Note: A proof that a discrete delta function is second-order accurate
Journal of Computational Physics
A convergence rate theorem for finite difference approximations to delta functions
Journal of Computational Physics
Superposition of Multi-Valued Solutions in High Frequency Wave Dynamics
Journal of Scientific Computing
Journal of Computational Physics
Finite difference methods for approximating Heaviside functions
Journal of Computational Physics
Discretizing delta functions via finite differences and gradient normalization
Journal of Computational Physics
High order numerical methods to two dimensional delta function integrals in level set methods
Journal of Computational Physics
High Order Numerical Methods to Three Dimensional Delta Function Integrals in Level Set Methods
SIAM Journal on Scientific Computing
Optimal reconstruction of material properties in complex multiphysics phenomena
Journal of Computational Physics
Hi-index | 31.49 |
This paper presents two new methods for discretizing a Dirac delta function which is concentrated on the zero level set of a smooth function u: R^n@?R. The function u is only known at the discrete set of points belonging to a regular mesh covering R^n. These two methods are used to approximate integrals over the manifold defined by the level set. Both methods are conceptually simple and easy to implement. We present the results of numerical experiments indicating that as the mesh size h goes to zero, the rate of convergence is at least O(h) for the first method, and O(h^2) for the second method. We perform a limited analysis of the proposed algorithms, including a proof of convergence for both methods.