Two methods for discretizing a delta function supported on a level set

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 int...

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Bibliographic Details
Published in:Journal of computational physics 2007-01, Vol.220 (2), p.915-931
Main Author: Towers, John D.
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Computational techniques
Convergence
Delta function
Discretization
Exact sciences and technology
Level set method
Manifolds
Mathematical analysis
Mathematical methods in physics
Mathematical models
Physics
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2006.05.037