Finite difference methods for approximating Heaviside functions. (English) Zbl 1171.65014

The paper presents two algorithms to evaluate the integral \(\mathcal{I}:=\int_\Omega f(\vec{x})d\vec{x}\), where \(\vec{x}\in \mathbb{R}^n\), \(\Omega=\{\vec{x}:\,u(\vec{x})>0\}\) and \(\partial\Omega=\{\vec{x}:\,u(\vec{x})=0\}\) is a compact manifold of codimension one. The method to be employed is motivated by the expression \(\mathcal{I}=\int_{\mathbb{R}^n} H(u(\vec{x}))f(\vec{x})d\vec{x}\), where \(H\) is the Heaviside function. The approach consists of approximating \(H\) by finite differencing its first few primitives, a technique already used by the author to approximate delta functions.
A brief presentation of the algorithms is given below. Let
\[ I(z)=\int_0^z H(\zeta)d\zeta\quad\text{and}\quad J(z)=\int_0^z I(\zeta)d\zeta. \]
The following relationships are derived:
\[ I(u)=\langle\nabla J(u),\nabla u \rangle /|\nabla u|^2,\quad H(u)=\langle\nabla I(u),\nabla u \rangle /|\nabla u|^2, \]
where \(\langle \cdot,\cdot \rangle\) stands for the inner product. By discretizing \(H(u)\) the one-step algorithm \(FDMH_1\) is obtained which converges at a rate of \(\mathcal{O}(h^2)\) when \(u\) is smooth enough. By discretizing both relationships the two-step algorithm \(FDMH_2\) is derived which can converge at a rate of \(\mathcal{O}(h^3)\). These results are validated by means of some numerical examples.


65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
41A25 Rate of convergence, degree of approximation
41A63 Multidimensional problems
Full Text: DOI


[1] Adalsteinsson, D.; Sethian, J., A fast level set method for propagating interfaces, J. comput. phys., 118, 269, (1995) · Zbl 0823.65137
[2] Burden, R.; Faires, J., Numerical analysis, (1989), PWS-Kent Publishing Company Boston · Zbl 0671.65001
[3] Dahlquist, G.; Björk, A., Numerical methods, (1974), Prentice-Hall Englewood Cliffs, New Jersey
[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] 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
[6] Min, C.; Gibou, F., Robust second-order accurate discretizations of the multi-dimensional heaviside and Dirac delta functions, J. comput. phys., 227, 9686-9695, (2008) · Zbl 1153.65014
[7] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2003), Springer-Verlag New York · Zbl 1026.76001
[8] 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
[9] 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
[10] Sethian, J.A., Level set methods and fast marching methods, (1999), Cambridge University Press Cambridge · Zbl 0929.65066
[11] Tornberg, A.K., Multi-dimensional quadrature of singular and discontinuous functions, Bit, 42, 644-669, (2002) · Zbl 1021.65010
[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] 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
[14] J.D. Towers, Discretizing delta functions via finite differences and gradient normalization, J. Comput. Phys. in press, doi:10.1016/j.jcp.2009.02.012. · Zbl 1167.65007
[15] Towers, J.D., A convergence rate theorem for finite difference approximations to delta functions, J. comput. phys., 227, 6591-6597, (2008) · Zbl 1155.65016
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