Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions

Journal of Computational Physics

Volumn 227, Issue 22, 2008, Pages 9686-9695

  Min, Chohong ^a   Gibou, Frédéric ^b  

^a Kyung Hee University   (South Korea)

^b University of California   (United States)

Author keywords

Dirac delta function; Heaviside function; Level set methods; Singular source term

Indexed keywords

NUMERICAL METHODS;

DIRAC DELTA FUNCTION; DISCRETIZATIONS; HEAVISIDE; HEAVISIDE'S FUNCTIONS; LEVEL SET METHOD; MULTI DIMENSIONAL; SECOND ORDERS; SINGULAR SOURCE TERM; SINGULAR SOURCES; SOURCE TERMS;

DELTA FUNCTIONS;

EID: 52749093964     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2008.07.021     Document Type: Article

References (27)

1. Aslam T. A partial differential equation approach to multidimensional extrapolation. J. Comput. Phys. 193 (2004) 349-355
2. Brackbill J.U., Kothe D.B., and Zemach C. A continuum method for modelling surface tension. J. Comput. Phys. 100 (1992) 335-353
3. D. Calhoun, P. Smereka. The numerical approximation of a delta function. Preprint , 2004.
4. Engquist B., Tornberg A.K., and Tsai R. Discretization of Dirac delta functions in level set methods. J. Comput. Phys. 207 (2005) 28-51
5. Enright D., Fedkiw R., Ferziger J., and Mitchell I. A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183 (2002) 83-116
6. Goodman J.E., and O'Rourke J. The handbook of discrete and computational geometry (1997), CRC Press LL
7. Karma A., and Rappel W.-J. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 57 (1997) 4323-4349
8. Kuhn H.W. Some combinational lemmas in topology. IBM J. Res. Develop. 4 (1960) 508-524
9. Mayo A. The fast solution of Poisson's and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21 (1984) 285-299
10. Min C. Simplicial isosurfacing in arbitrary dimension and codimension. J. Comput. Phys. 190 (2003) 295-310
11. Min C. Local level set method in high dimension and codimension. J. Comput. Phys. 200 (2004) 368-382
12. Min C., and Gibou F. Geometric integration over irregular domains with application to level set methods. J. Comput. Phys. 226 (2007) 1432-1443
13. Min C., and Gibou F. A second order accurate level set method on non-graded adaptive Cartesian grids. J. Comput. Phys. 225 (2007) 300-321
14. Osher S., and Fedkiw R. Level Set Methods and Dynamic Implicit Surfaces (2002), Springer-Verlag, New York, NY
15. Peskin C. The immersed boundary method. Acta Numer. 11 (2002) 479-517
16. Sallee J.F. The middle-cut triangulations of the n-cube. SIAM J. Alg. Disc. Methods 5 (1984) 407-419
17. Sethian J. Fast marching methods. SIAM Rev. 41 (1999) 199-235
18. Sethian J.A. Level Set Methods and Fast Marching Methods (1999), Cambridge University Press, Cambridge
19. Smereka P. Semi-implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput. 19 (2003) 439-456
20. Smereka P. The numerical approximation of a delta function with application to level set methods. J. Comput. Phys. 211 (2006) 77-90
21. Sussman M., and Puckett E.G. A coupled level let and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162 (2000) 301-337
22. Sussman M., Smereka P., and Osher S. A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1994) 146-159
23. Tornberg A.-K., and Engquist B. Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200 (2004) 462-488
24. Towers J. Two methods for discretizing a delta function supported on a level set. J. Comput. Phys. 220 (2007) 915-931
25. Towers J. A convergence rate theorem for finite difference approximations to delta functions. J. Comput. Phys. 227 (2008) 6591-6597
26. Tryggvason G., Bunner B., Esmaeeli A., Juric D., Al-Rawahi N., Tauber W., Han J., Nas S., and Jan Y.-J. A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169 (2001) 708-759
27. Walén J. On the approximation of singular source terms in differential equations. Numer. Methods Partial Differ. Equat. 15 (1999) 503-520