The numerical approximation of a delta function with application to level set methods. (English) Zbl 1086.65503

Summary: It is shown that a discrete delta function can be constructed using a technique developed by A. Mayo [SIAM J. Numer. Anal. 21, No. 2, 285–299 (1984)] for the numerical solution of elliptic equations with discontinuous source terms. This delta function is concentrated on the zero level set of a continuous function. In two space dimensions, this corresponds to a line and a surface in three space dimensions. Delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function. The numerical implementation of these delta functions achieves the expected order of accuracy.


65D15 Algorithms for approximation of functions
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N06 Finite difference methods for boundary value problems involving PDEs
Full Text: DOI


[1] Beyer, R.P.; Leveque, R.J., Analysis of a one-dimensional model for the immersed boundary method, SIAM J. numer. anal., 29, 332-364, (1992) · Zbl 0762.65052
[2] D. Calhoun, P. Smereka, The numerical approximation of a delta function, preprint (2004). Available from: <http://www.math.lsa.umich.edu/ psmereka/notes.html>.
[3] B. Engquist, A.K. Tornberg, R. Tsai, Discretization of Dirac delta functions in level set methods, UCLA CAM Report, 2004. · Zbl 1074.65025
[4] Hou, T.Y.; Li, Z.L.; Osher, S.; Zhao, H.K., A hybrid method for moving interface problems with application to the Hele-Shaw flow, J. comput. phys., 134, 236-252, (1997) · Zbl 0888.76067
[5] LeVeque, R.J.; Li, Z.L., Immersed interface methods for Stokes flow with elastic boundaries, SIAM J. sci. comput., 18, 709-735, (1997) · Zbl 0879.76061
[6] Liu, X.D.; Fedkiw, R.; Kang, M., A boundary condition capturing method for poisson’s equation on irregular domains, J. comput. phys., 160, 151-178, (2000) · Zbl 0958.65105
[7] Mayo, A., The fast solution of poisson’s and the biharmonic equations on irregular regions, SIAM J. sci. comput., 21, 285-299, (1984) · Zbl 1131.65303
[8] Mayo, A., The rapid evaluation of volume integrals of potential theory on general regions, J. comput. phys., 100, 236-245, (1992) · Zbl 0772.65012
[9] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2000), Springer New York
[10] Peng, D.; Merriman, B.; Osher, S.J.; Zhao, H.K.; Kang, M.J., A PDE-based fast local level set method, J. comput. phys., 155, 410-438, (1999) · Zbl 0964.76069
[11] Sussman, M.; Smereka, P.; Osher, S., A level set method for computing solutions to incompressible two phase flow, J. comput. phys., 119, 146-159, (1994) · Zbl 0808.76077
[12] Tornberg, A.K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392
[13] Unverdi, S.O.; Trygvasson, G., A front-tracking method for the computation of multiphase flow, J. comput. phys., 100, 25-37, (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.