Essentially non-oscillatory adaptive tree methods

Journal of Scientific Computing

Volumn 35, Issue 1, 2008, Pages 25-41

  Cecil, Thomas C ^a   Osher, Stanley J ^b   Qian, Jianliang ^c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Adaptive tree methods; Essentially non oscillatory; Hamilton Jacobi equations

Indexed keywords

ADAPTIVE STRATEGIES; ADAPTIVE TREE; BUSINESS MEDIA; ESSENTIALLY NON OSCILLATORY; HIGH ORDER; NEW METHODS; NUMERICAL EXAMPLES; SPRINGER (CO); TREE STRUCTURES; UNSTRUCTURED DATA;

BINARY TREES; DATA STRUCTURES; FILE ORGANIZATION; NUMBER THEORY; TREES (MATHEMATICS);

NUMERICAL METHODS;

EID: 44849094864     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-007-9164-6     Document Type: Article

References (30)

1. Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 114, 45-58 (1994)
2. Albert, S., Cockburn, B., French, D., Peterson, T.: A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part I: The steady state case. Math. Comput. 71, 49-76 (2002)
3. Berger, M., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484-512 (1984)
4. Berger, M.J., LeVeque, R.J.: Adaptive mesh refinement using wave propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35, 2298-2316 (1998)
5. Casper, J., Atkins, H.L.: A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems. J. Comput. Phys. 106, 62-76 (1993)
6. Cecil, T., Osher, S., Qian, J.: Simplex free adaptive tree fast sweeping and evolution methods for solving level set equations in arbitrary dimension. J. Comput. Phys. 213(2), 458-473 (2006)
7. Cockburn, B., Yenikaya, B.: An adaptive method with rigorous error control for the Hamilton-Jacobi equations. Part I: The one dimensional steady state case. Appl. Numer. Math. 52, 175-195 (2005)
8. Frisken, S.F., Perry R.N.: Simple and efficient traversal methods for quadtrees and octrees. J. Graphics Tools 7(3), 1-11 (2002)
9. Glimm, J., Grove, J., Li, X., Oh, W., Tan, D.C.: The dynamics of bubble growth for Rayleigh-Taylor unstable interfaces. Phys. Fluids 31, 174-201 (1988)
10. Gosse, L., Makridakis, C.: Two a posteriori error estimates for one-dimensional scalar conservation laws. SIAM J. Numer. Anal. 38, 964-988 (2000)
11. Harten, A., Engquist, B., Osher, S.J., Chakravarthy, S.: Uniformly high order essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231-303 (1987)
12. Jiang, G.S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126-2143 (2000)
13. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202-228 (1996)
14. Knuth, D.E.: The Art of Computer Programming, vol. 1: Fundamental algorithms, 3rd edn. Addison-Wesley, Redwood City (1997)
15. Kroner, D., Ohlberger, M.: A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions. Math. Comp. 69, 25-39 (2000)
16. LeVeque, R.J.: Numerical methods for conservation laws. Birkhauser, Basel (1990)
17. Merriman, B.: Understanding the Shu-Osher conservative finite difference form. J. Sci. Comput. 19, 309-322 (2003)
18. Min, C: Local level set method in high dimension and codimension. J. Comput. Phys. 200(1), 368-382 (2004)
19. Moore, D.: The cost of balancing generalized quadtrees. In: SMA '95: Proceedings of the Third ACM Symposium on Solid Modeling and Applications, pp. 305-312. ACM, New York (1995)
20. Osher, S.J., Shu, C.W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28, 907-922 (1991)
21. Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Oxford University Press, London (2002)
22. Qian, J., Symes, W.W.: Adaptive finite difference method for traveltime and amplitude. Geophysics 67, 167-176 (2002)
23. Remacle, J.-F., Flaherty, J.E., Shephard, M.S.: An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev. 45, 53-72 (2003)
24. Samet, H.: Applications of Spatial Data Structures: Computer Graphics, Image Processing, and GIS. Addison-Wesley, Boston (1990)
25. Serna, S., Qian, J.: Fifth order weighted power-ENO schemes for Hamilton-Jacobi equations. J. Sci. Comput. 29, 57-81 (2006)
26. Shi, J., Zhang, Y.-T., Shu, C.-W.: Resolution of high order WENO schemes for complicated flow structures. J. Comput. Phys. 186, 690-696 (2003)
27. Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Cockburn, B., Johnson, C., Shu, C.W., Tadmor, E. (eds.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697, pp. 325-432. Springer, Berlin (1998)
28. Shu, C.W., Osher, S.J.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439-471 (1988)
29. Shu, C.W., Osher, S.J.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32-78 (1989)
30. Strain, J.: Tree methods for moving interfaces. J. Comput. Phys. 151(2), 616-648 (1999)