##
**Discretization of Dirac delta functions in level set methods.**
*(English)*
Zbl 1074.65025

The authors analyze the accuracy of regularizations of Dirac delta functions in the context of the level set method. Following a recent work by A. K. Tornberg and B. Engquist [J. Comput Phys. 200, No. 2, 462–488 (2004; Zbl 1115.76392)] showing that the most common technique for the regularization of the delta function in level set methods is inconsistent and may lead to \(O(1)\) errors, the authors present two techniques to construct consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both techniques are based solely on the distance to the singularity and thus are independent of the grid. The first of them is based on a tensor product of regularized one-dimensional delta functions.

In the second technique a variable support of the regularization domain is used. The regularization is constructed. from a one-dimensional regularization that is extended to multi-dimensions. Being the Dirac delta function \(\delta\) previously replaced by a more regular function \(\delta_\varepsilon\), the multidimensional regularized delta function is then defined as \(\delta_\varepsilon(\Gamma, x)= \delta_\varepsilon(d(\Gamma, x))\) where \(d(\Gamma, x)\) stands for the distance function. Convergence analysis, numerical results and some applications to a class of partial differential equations are also given.

In the second technique a variable support of the regularization domain is used. The regularization is constructed. from a one-dimensional regularization that is extended to multi-dimensions. Being the Dirac delta function \(\delta\) previously replaced by a more regular function \(\delta_\varepsilon\), the multidimensional regularized delta function is then defined as \(\delta_\varepsilon(\Gamma, x)= \delta_\varepsilon(d(\Gamma, x))\) where \(d(\Gamma, x)\) stands for the distance function. Convergence analysis, numerical results and some applications to a class of partial differential equations are also given.

Reviewer: Nácere Hayek (La Laguna)

### MSC:

65D20 | Computation of special functions and constants, construction of tables |

46F10 | Operations with distributions and generalized functions |

### Keywords:

discretization of Dirac delta functions; level set methods; consistent approximations; regularized delta functions; regularization; convergence analysis; numerical results### Citations:

Zbl 1115.76392
PDF
BibTeX
XML
Cite

\textit{B. Engquist} et al., J. Comput. Phys. 207, No. 1, 28--51 (2005; Zbl 1074.65025)

Full Text:
DOI

### References:

[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] | B. Burger, S. Osher, A survey on level set methods for inverse problems and optimal design, UCLA CAM Report No 04-02, 2004 |

[3] | Cheng, L.-T., Construction of shapes arising from the Minkowski problem using a level set approach, J. sci. comput., 19, 123-138, (2003) · Zbl 1035.65093 |

[4] | Li-Tien Cheng, The level set method applied to geometrically based motion, materials science, and image processing, PhD thesis, UCLA, 2000 |

[5] | Fraleigh, J.B., A first course in abstract algebra, (1989), Addison-Wesley Publishing Co. · Zbl 0697.00001 |

[6] | LeVeque, Randall J.; Li, Zhi Lin, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 1019-1044, (1994) · Zbl 0811.65083 |

[7] | Liu, X.-D.; Fedkiw, R.; Kang, M., A boundary condition capturing method for poisson’s equation on irregular domains, J. comput. phys., 160, 1, 151-178, (2000) · Zbl 0958.65105 |

[8] | Osher, S.J.; Fedkiw, R.P., Level set methods and dynamic implicit surfaces, (2002), Springer |

[9] | Peskin, C.S., The immersed boundary method, Acta numer., 11, 479-517, (2002) · Zbl 1123.74309 |

[10] | Sethian, J.A., Level set methods and fast marching methods. evolving interfaces in computational geometry, fluid mechanics, computer vision and materials science, (1999), Cambridge University Press · Zbl 0973.76003 |

[11] | Sussman, M.; Smereka, P.; Osher, S., A level set method for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146-159, (1994) · Zbl 0808.76077 |

[12] | Tornberg, A.K., Multi-dimensional quadrature of singular and discontinuous functions, Bit, 42, 644-669, (2002) · Zbl 1021.65010 |

[13] | Tornberg, A.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392 |

[14] | Tornberg, A.K.; Engquist, B., Regularization techniques for numerical approximation of PDEs with singularities, J. sci. comput., 19, 527-552, (2003) · Zbl 1035.65085 |

[15] | Tornberg, A.K.; Engquist, B., The segment projection method for interface tracking, Commun. pure appl. math., 56, 47-79, (2003) · Zbl 1205.76205 |

[16] | 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 |

[17] | Waldén, J., On the approximation of singular source terms in differential equations, Numer. meth. part. D E, 15, 503-520, (1999) · Zbl 0938.65112 |

[18] | Zhao, H.-K.; Chan, T.; Merriman, B.; Osher, S., A variational level set approach to multiphase motion, J. comput. phys., 127, 179-195, (1996) · Zbl 0860.65050 |

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.