Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions. (English) Zbl 1153.65014

Summary: We present a robust second-order accurate method for discretizing the multi-dimensional Heaviside and the Dirac delta functions on irregular domains. The method is robust in the following ways: (1) integrations of source terms on a co-dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain’s boundary; (2) the method depends only on the function value of a level function, not on its derivatives.
We present the discretizations in tabulated form to make their implementations straightforward. We present numerical results in two and three spatial dimensions to demonstrate the second-order accuracy in the \(L^{1}\)-norm in the case of the solution of partial differential equations with singular source terms. In the case of evaluating the contribution of singular source terms on interfaces, the method is also second-order accurate in the \(L^{\infty }\)-norm.


65D15 Algorithms for approximation of functions
Full Text: DOI


[1] Aslam, T., A partial differential equation approach to multidimensional extrapolation, J. comput. phys., 193, 349-355, (2004) · Zbl 1036.65002
[2] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modelling surface tension, J. comput. phys., 100, 335-353, (1992) · Zbl 0775.76110
[3] D. Calhoun, P. Smereka. The numerical approximation of a delta function. Preprint <http://www.math.lsa.umich.edu/ psmereka/notes.html>, 2004.
[4] Engquist, B.; Tornberg, A.K.; Tsai, R., Discretization of Dirac delta functions in level set methods, J. comput. phys., 207, 28-51, (2005) · Zbl 1074.65025
[5] Enright, D.; Fedkiw, R.; Ferziger, J.; Mitchell, I., A hybrid particle level set method for improved interface capturing, J. comput. phys., 183, 83-116, (2002) · Zbl 1021.76044
[6] Goodman, J.E.; O’Rourke, J., The handbook of discrete and computational geometry, (1997), CRC Press LL · Zbl 0890.52001
[7] Karma, A.; Rappel, W.-J., Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. rev. E, 57, 4323-4349, (1997) · Zbl 1086.82558
[8] Kuhn, H.W., Some combinational lemmas in topology, IBM J. res. develop., 4, 508-524, (1960) · Zbl 0109.15603
[9] Mayo, A., The fast solution of poisson’s and the biharmonic equations on irregular regions, SIAM J. numer. anal., 21, 285-299, (1984) · Zbl 1131.65303
[10] Min, C., Simplicial isosurfacing in arbitrary dimension and codimension, J. comput. phys., 190, 295-310, (2003) · Zbl 1029.65016
[11] Min, C., Local level set method in high dimension and codimension, J. comput. phys., 200, 368-382, (2004) · Zbl 1086.65088
[12] Min, C.; Gibou, F., Geometric integration over irregular domains with application to level set methods, J. comput. phys., 226, 1432-1443, (2007) · Zbl 1125.65021
[13] Min, C.; Gibou, F., A second order accurate level set method on non-graded adaptive Cartesian grids, J. comput. phys., 225, 300-321, (2007) · Zbl 1122.65077
[14] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2002), Springer-Verlag New York, NY
[15] Peskin, C., The immersed boundary method, Acta numer., 11, 479-517, (2002) · Zbl 1123.74309
[16] Sallee, J.F., The middle-cut triangulations of the n-cube, SIAM J. alg. disc. methods, 5, 407-419, (1984) · Zbl 0543.52004
[17] Sethian, J., Fast marching methods, SIAM rev., 41, 199-235, (1999) · Zbl 0926.65106
[18] Sethian, J.A., Level set methods and fast marching methods, (1999), Cambridge University Press Cambridge · Zbl 0929.65066
[19] Smereka, P., Semi-implicit level set methods for curvature and surface diffusion motion, J. sci. comput., 19, 439-456, (2003) · Zbl 1035.65098
[20] Smereka, P., The numerical approximation of a delta function with application to level set methods, J. comput. phys., 211, 77-90, (2006) · Zbl 1086.65503
[21] Sussman, M.; Puckett, E.G., A coupled level let and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows, J. comput. phys., 162, 301-337, (2000) · Zbl 0977.76071
[22] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146-159, (1994) · Zbl 0808.76077
[23] Tornberg, A.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392
[24] Towers, J., Two methods for discretizing a delta function supported on a level set, J. comput. phys., 220, 915-931, (2007) · Zbl 1115.65028
[25] Towers, J., A convergence rate theorem for finite difference approximations to delta functions, J. comput. phys., 227, 6591-6597, (2008) · Zbl 1155.65016
[26] Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J., A front-tracking method for the computations of multiphase flow, J. comput. phys., 169, 708-759, (2001) · Zbl 1047.76574
[27] Walén, J., On the approximation of singular source terms in differential equations, Numer. methods partial differ. equat., 15, 503-520, (1999)
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.