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Discretizing delta functions via finite differences and gradient normalization. (English) Zbl 1167.65007
The aim of this paper is to create discretizations of certain integrals involving \(\delta\)-functions in the integrand, especially in dimension \(n\) at most three. Two special types of integrals are considered, namely one that has products of \(\delta\)-functions with norms of wedge products of gradients as integrands, and another one which has products of \(\delta\)-functions and functions of \(n\)-variables inside the integral. For the purpose of discretizing these objects, finite difference approaches are used. Many numerical examples are offered to show the usefulness of the ideas, where of course the accuracy of the discretizations is of special interest.

65D15 Algorithms for approximation of functions
46F10 Operations with distributions and generalized functions
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[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] P. Burchard, L.-T. Cheng, B. Merriman, S. Osher, Motion of curves in three spatial dimensions using a level set approach, 2001. <http://www.math.ucla.edu/applied/cam/index.html>. · Zbl 0991.65077
[3] Cheng, L.-T.; Burchard, P.; Merriman, B.; Osher, S., Motion of curves constrained on surfaces using a level set approach, J. comput. phys., 175, 2, 604-644, (2002) · Zbl 0996.65013
[4] Cheng, L.-T.; Liu, H.; Osher, S., Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations, Commun. math. sci., 1, 593-621, (2003) · Zbl 1084.35066
[5] 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
[6] Flanders, H., Differential forms with applications to the physical sciences, (1989), Dover Publications New York · Zbl 0733.53002
[7] Jin, S.; Osher, S., A level set method for the computation of multivalued solutions to quasilinear hyperbolic PDE’s and hamilton – jacobi equations, Commun. math. sci., 1, 3, 575-591, (2003) · Zbl 1090.35116
[8] Jin, S.; Liu, H.; Osher, S.; Tsai, R., Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation, J. comput. phys., 205, 222-241, (2005) · Zbl 1072.65132
[9] Liu, H.; Osher, S.; Tsai, R., Multi-valued solution and level set methods in computational high frequency wave propagation, Commun. comput. phys., 1, 5, 765-804, (2006) · Zbl 1120.65110
[10] C. Misner, Spherical harmonic decomposition on a cubic grid, 1999. arxiv:gr-qc/9910044 v1.
[11] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2003), Springer-Verlag New York · Zbl 1026.76001
[12] Osher, S.; Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132
[13] Peng, D.; Merriman, B.; Osher, S.; Zhao, H.; Kang, M., A PDE-based fast local level set method, J. comput. phys., 155, 410-438, (1999) · Zbl 0964.76069
[14] Runborg, O., Mathematical models and numerical methods for high frequency waves, Commun. comput. phys., 2, 5, 827-880, (2007) · Zbl 1164.78300
[15] Sethian, J.A., Level set methods and fast marching methods, (1999), Cambridge University Press Cambridge · Zbl 0929.65066
[16] 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
[17] Tornberg, A.K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392
[18] Tornberg, A.K.; Engquist, B., Regularization techniques for numerical approximation of PDEs with singularities, J. sci. comput., 19, 527-552, (2003) · Zbl 1035.65085
[19] 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
[20] Zhao, H.K.; Chan, T.; Merriman, B.; Osher, S., Variational level set approach to multiphase motion, J. comput. phys., 127, 179-195, (1996) · Zbl 0860.65050
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