High order numerical methods to three dimensional delta function integrals in level set methods

SIAM Journal on Scientific Computing

Volumn 32, Issue 3, 2010, Pages 1288-1309

  Wen, Xin ^a  

^a CHINESE ACADEMY OF SCIENCES   (China)

Author keywords

Delta function integral; High order numerical method; Level set method

Indexed keywords

CELLS; CYTOLOGY; DELTA FUNCTIONS; DROP BREAKUP; HYDROGELS; INTEGRAL EQUATIONS; LEVEL MEASUREMENT; MESH GENERATION;

DELTA FUNCTION INTEGRAL; ERROR CANCELLATION; HIGH-ORDER METHODS; HIGH-ORDER NUMERICAL METHOD; HIGHER ORDER ACCURACY; HIGHER-ORDER METHODS; LEVEL SET METHOD; MESH CELLS; ONE-DIMENSIONAL; TWO-DIMENSIONAL;

NUMERICAL METHODS;

EID: 77953827201     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/090758295     Document Type: Article

References (47)

1. D. Adalsteinsson and J. A. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys., 118 (1995), pp. 269-277.
2. J. T. Beale, A proof that a discrete delta function is second-order accurate, J. Comput. Phys., 227 (2008), pp. 2195-2197.
3. P. Burchard, L.-T. Cheng, B. Merriman, and S. Osher, Motion of Curves in Three Spatial Dimensions Using a Level Set Approach, J. Comput. Phys., 170 (2001), pp. 720-741.
4. J. T. Beale and A. Majda, Vortex methods. II: Higher order accuracy in two and three dimensions, Math. Comp., 39 (1982), pp. 29-52.
5. M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16 (2005), pp. 263-301.
6. L.-T. Cheng, Construction of shapes arising from the Minkowski problem using a level set approach, J. Sci. Comput., 19 (2003), pp. 123-138.
7. A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57 (1973), pp. 785-796.
8. B. Engquist, A. K. Tornberg, and R. Tsai, Discretization of Dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), pp. 28-51.
9. S. Jin and X. Liao, A Hamiltonian-preserving scheme for high frequency elastic waves in heterogeneous media, J. Hyperbolic Differ. Equ. 3 (2006), pp. 741-777.
10. S. Jin, H. L. Liu, S. Osher, and R. Tsai, Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation, J. Comput. Phys., 205 (2005), pp. 222-241.
11. S. Jin, H. L. Liu, S. Osher, and R. Tsai, Computing multi-valued physical observables for high frequency limit of symmetric hyperbolic systems, J. Comput. Phys., 210 (2005), pp. 497-518.
12. S. Jin and X. Wen, Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials, Commun. Math. Sci., 3 (2005), pp. 285-315.
13. S. Jin and X. Wen, Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds, J. Comput. Phys., 214 (2006), pp. 672-697.
14. S. Jin and X. Wen, A Hamiltonian-preserving scheme for the Liouville equation of geometrical optics with partial transmissions and reflections, SIAM J. Numer. Anal., 44 (2006), pp. 1801-1828. (Pubitemid 350012041)
15. S. Jin, H. Wu, and Z. Huang, A hybrid phase-flow method for Hamiltonian systems with discontinuous Hamiltonians, SIAM J. Sci. Comput., 31 (2008), pp. 1303-1321.
16. S. Jin, H.Wu, and X. Yang, Gaussian beam methods for the Schrödinger equation in the semiclassical regime: Lagrangian and Eulerian formulations, Commun. Math. Sci., 6 (2008), pp. 995-1020.
17. S. Jin, H. Wu, and X. Yang, A numerical study of the Gaussian beam methods for onedimensional Schrödinger-Poisson equations, J. Comput. Math., 28 (2010), pp. 261-272.
18. S. Jin, H. Wu, X. Yang, and Z. Huang, Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials, J. Comput. Phys., to appear.
19. S. Jin and X. Yang, Computation of the semiclassical limit of the Schrödinger equation with phase shift by a level set method, J. Sci. Comput., 35 (2008), pp. 144-169.
20. S. Jin and D. Yin, Computational high frequency waves through curved interfaces via the Loiuville equation and geometric theory of diffraction, J. Comput. Phys., 227 (2008), pp. 6106-6139.
21. S. Jin and D. Yin, Computation of high frequency wave diffraction by a half plane via the Loiuville equation and geometric theory of diffraction, Commun. Comput. Phys., 4 (2008), pp. 1106-1128.
22. D. Juric and G. Tryggvason, A front-tracking method for dendritic solidification, J. Comput. Phys., 123 (1996), pp. 127-148.
23. S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semiclassical regime, J. Comput. Phys., 228 (2009), pp. 2951-2977.
24. C. Min, Local level set method in high dimension and codimension, J. Comput. Phys., 200 (2004), pp. 368-382.
25. C. Min and F. Gibou, Geometric integration over irregular domains with application to levelset methods, J. Comput. Phys., 226 (2007), pp. 1432-1443.
26. C. Min and F. Gibou, Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions, J. Comput. Phys., 227 (2008), pp. 9686-9695.
27. S. Osher, L. T. Cheng, M. Kang, H. Shim, and Y. H. Tsai, Geometric optics in a phase space based level set and Eulerian framework, J. Comput. Phys., 179 (2002), pp. 622-648.
28. S. Osher and R. P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag, New York, 2002.
29. D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, A PDE-based fast local level set method, J. Comput. Phys., 155 (1999), pp. 410-438.
30. C. S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), pp. 220-252.
31. C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), pp. 479-511.
32. J. A. Sethian, Level Set Methods and Fast Marching Methods, Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science. Cambridge University Press, Cambridge, 1999.
33. P. Smereka, The numerical approximation of a delta function with application to level set methods, J. Comput. Phys., 211 (2006), pp. 77-90.
34. M. Sussman, P. Smereka, and S. Osher, A level set method for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), pp. 146-159.
35. A.-K. Tornberg, Interface Tracking Methods with Applications to Multiphase Flows, Ph.D. thesis, Royal Institute of Technology, Stockholm, Sweden, 2000.
36. A.-K. Tornberg and B. Engquist, Regularization techniques for numerical approximation of PDEs with singularities, J. Sci. Comput., 19 (2003), pp. 527-552.
37. A.-K. Tornberg and B. Engquist, Numerical approximations of singular source terms in differential equations, J. Comput. Phys., 200 (2004), pp. 462-488.
38. J. D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys., 220 (2007), pp. 915-931.
39. J. D. Towers, A convergence rate theorem for finite difference approximations to delta functions, J. Comput. Phys., 227 (2008), pp. 6591-6597.
40. G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas, and Y. J. Jan, A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169 (2001), pp. 708-759.
41. S. O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100 (1992), pp. 25-37.
42. G. W. Wei, Y. B. Zhao, and Y. Xiang, Discrete singular convolution and its application to the analysis of plates with internal supports, Part 1: Theory and algorithm, Internat. J. Numer. Methods Engrg., 55 (2002), pp. 913-946. (Pubitemid 35206179)
43. X. Wen, High order numerical methods to a type of delta function integrals, J. Comput. Phys., 226 (2007), pp. 1952-1967.
44. X. Wen, High order numerical quadratures to one dimensional delta function integrals, SIAM J. Sci. Comput., 30 (2008), pp. 1825-1846.
45. X. Wen, High order numerical methods to two dimensional delta function integrals in level set methods, J. Comput. Phys., 228 (2009), pp. 4273-4290.
46. X. Wen, A high order numerical method for computing physical observables in the semiclassical limit of the one dimensional linear Schrödinger equation with discontinuous potentials, J. Sci. Comput., 42 (2010), pp. 318-344.
47. H.-K. Zhao, T. Chan, B. Merriman, and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), pp. 179-195.