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Two methods for discretizing a delta function supported on a level set. (English) Zbl 1115.65028
Let $$f:{\mathbb R}^n \to {\mathbb R}$$ and $$u:{\mathbb R}^n \to {\mathbb R}$$ be smooth functions which are given by their data on a grid. Let $$\Gamma$$ be the zero level set of $$u$$. The author considers the problem of approximating the integral $$\int_{\Gamma} f(x)\,ds$$. It is common practice to replace the integral above by $\int_{{\mathbb R}^n} f(x)\, \delta(u(x))\,\| \nabla u(x)\| \, dx,$ where $$\delta$$ denotes the Dirac delta function. Then one approximates this integral using the available grid-defined function values. The author proposes two methods for discretization of $$\delta(u(x))$$.

##### MSC:
 65D32 Numerical quadrature and cubature formulas 58C35 Integration on manifolds; measures on manifolds 46F10 Operations with distributions and generalized functions 41A55 Approximate quadratures
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##### References:
  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  B. Engquist, A.K. Tornberg, R. Tsai, Discretization of Dirac Delta Functions in Level Set Methods. Preprint available from: . · Zbl 1074.65025  Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2003), Springer-Verlag New York · Zbl 1026.76001  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  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  Peskin, C.S., The immersed boundary method, Acta numerica, 11, 479-511, (2002) · Zbl 1123.74309  Sethian, J.A., Level set methods and fast marching methods, (1999), Cambridge University Press Cambridge · Zbl 0929.65066  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  Spivak, M., Calculus on manifolds, (1965), W.A. Benjamin, Inc. New York · Zbl 0141.05403  Tornberg, A.K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392  Tornberg, A.K.; Engquist, B., Regularization techniques for numerical approximation of PDEs with singularities, J. sci. comput., 19, 527-552, (2003) · Zbl 1035.65085  H.K. Zhao, S. Osher, T. Chan, B. Merriman, Variational formulation for motion of multiple junctions and interfaces by level set approach, Preprint available from: .
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