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Delta function approximations in level set methods by distance function extension. (English) Zbl 1186.65018
Summary: A.-K. Tornberg and B. Engquist [ibid. 200, No. 2, 462–488 (2004; Zbl 1115.76392)], it was shown for simple examples that the most common way to regularize delta functions in connection to level set methods produces inconsistent approximations with errors that are not reduced with grid refinement. Since then, several clever approximations have been derived to overcome this problem. However, the great appeal of the old method was its simplicity.
In this paper it is shown that the old method – a one-dimensional delta function approximation extended to higher dimensions by a distance function – can be made accurate with a different class of one-dimensional delta function approximations. The prize to pay is a wider support of the resulting delta function approximations.

MSC:
65D15 Algorithms for approximation of functions
Software:
Mulprec
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[1] Osher, S.; Sethian, J.A., Fronts propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132
[2] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2003), Springer Verlag · Zbl 1026.76001
[3] Tornberg, A.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392
[4] Peskin, C.S., The immersed boundary method, Acta numer., 11, 479-517, (2002) · Zbl 1123.74309
[5] Sethian, J., Level set methods and fast marching methods, (1999), Cambridge University Press · Zbl 0929.65066
[6] 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
[7] 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
[8] Beale, J.T., A proof that a discrete delta function is second-order accurate, J. comput. phys., 227, 2195-2197, (2008) · Zbl 1136.65017
[9] Towers, J.D., Two methods for discretizing a delta function supported on a level set, J. comput. phys., 220, 915-931, (2007) · Zbl 1115.65028
[10] Towers, J.D., A convergence rate theorem for finite difference approximations to delta functions, J. comput. phys., 227, 6591-6597, (2008) · Zbl 1155.65016
[11] Tornberg, A.-K.; Engquist, B., Regularization techniques for numerical approximation of pdes with singularities, J. scient. comp., 19, 527-552, (2003) · Zbl 1035.65085
[12] Tornberg, A.-K., Multi-dimensional quadrature of singular and discontinuous functions, Bit, 42, 644-669, (2002) · Zbl 1021.65010
[13] Olsson, E.; Kreiss, G., A conservative level set method for two phase flow, J. comput. phys., 210, 225-246, (2005) · Zbl 1154.76368
[14] Olsson, E.; Kreiss, G.; Zahedi, S., A conservative level set method for two phase flow II, J. comput. phys., 225, 785-807, (2007) · Zbl 1256.76052
[15] Beyer, R.P.; Leveque, R.J., Analysis of a one-dimensional model for the immersed boundary method, SIAM J. numer. anal., 29, 2, 332-364, (1992) · Zbl 0762.65052
[16] B.E. Segee, Using spectral techniques for improved performance in artificial neural networks, pp. 500-505.
[17] Dahlquist, G.; Björck, Å., Numerical methods in scientific computing, (2008), SIAM · Zbl 1153.65001
[18] Davis, P.J.; Rabinowitz, P., Numerical integration, (1967), Blaisdel Publishing Company · Zbl 0154.17802
[19] Munkres, J.R., Analysis on manifolds, (1991), Westview Press · Zbl 0743.26006
[20] Munkres, J.R., Elementary differential topology, (1963), Princeton University Press · Zbl 0107.17201
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