An accurate spectral/discontinuous finite-element formulation of a phase-space-based level set approach to geometrical optics

Journal of Computational Physics

Volumn 208, Issue 1, 2005, Pages 175-195

  Cockburn, Bernardo ^a   Qian, Jianliang ^b   Reitich, Fernando ^a   Wang, Jing ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Discontinuous Galerkin; Eikonal equation; Geometrical optics; Liouville equation; Spectral methods; Wave equation

Indexed keywords

FINITE ELEMENT METHOD; GALERKIN METHODS; GEOMETRY; LIOUVILLE EQUATION; NUMERICAL METHODS; PHASE SPACE METHODS; RUNGE KUTTA METHODS;

DISCONTINUOUS FINITE ELEMENT; DISCONTINUOUS GALERKIN; EIKONAL EQUATION; FINITE ELEMENT FORMULATIONS; HIGH-ORDER; HIGHER-ORDER; LIOUVILLE'S EQUATION; PHASE SPACES; SPACE-BASED; SPECTRAL METHODS;

GEOMETRICAL OPTICS;

EID: 19044395637     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jcp.2005.02.009     Document Type: Article

References (38)

1. R. Abgrall J.-D. Benamou Big ray tracing and eikonal solver on unstructured grids: Application to the computation of a multivalued traveltime field in the Marmousi model Geophysics 64 1999 230-239
2. D. Andersh J. Moore S. Kosanovich D. Kapp R. Bhalla R. Kipp T. Courtney A. Nolan F. German J. Cook J. Hughes Xpatch 4: The next generation in high frequency electromagnetic modeling and simulation software IEEE Natl. Radar Conf. Proc. 2000 844-849
3. E. Bécache P. Joly C. Tsogka Fictitious domains, mixed finite elements and perfectly matched layers for 2-D elastic wave propagation J. Comput. Acoust. 9 2001 1175-1201
4. J.-D. Benamou Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations Comm. Pure Appl. Math. 52 1999 1443-1475
5. E. Bleszynski M. Bleszynski T. Jaroszewicz AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems Radio Sci. 31 1996 1225-1251
6. B. Borden Radar Imaging of Airborne Targets 1999 Institute of Physics Publishing Bristol
7. M. Born E. Wolf Principles of Optics 1980 Cambridge University Press Cambridge
8. O.P. Bruno C.A. Geuzaine J.A. Monroe F. Reitich Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: The convex case Phil. Trans. Roy. Soc. Lond. 362 2004 629-645
9. O. Bruno L. Kunyansky A fast, high-order algorithm for the solution of surface scattering problems: Basic implementation tests, and application J. Comput. Phys. 169 2001 80-110
10. M.-H. Chen, B. Cockburn, F. Reitich, High-order RKDG methods for computational electromagnetics, J. Sci. Comput. (in press)
11. L.-T. Cheng M. Kang S. Osher H. Shim Y.-H. Tsai Reflection in a level set framework for geometric optics CMES Comput. Meth. Sci. Eng. 5 2004 347-360
12. Jon F. Claerbout Imaging the Earth's Interior 1985 Blackwell Scientific Publications Cambridge, MA
13. B. Cockburn C.-W. Shu The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems J. Comput. Phys. 141 1998 199-224
14. B. Cockburn C.-W. Shu Runge-Kutta discontinuous Galerkin methods for convection-dominated problems J. Sci. Comput. 16 2001 173-261
15. M.G. Crandall P.-L. Lions Viscosity solutions of Hamilton-Jacobi equations Trans. Am. Math. Soc. 277 1983 1-42
16. B. Engquist O. Runborg Computational high frequency wave propagation Acta Numer. 2003 181-266
17. B. Engquist O. Runborg A.-K. Tornberg High-frequency wave propagation by the segment projection method J. Comput. Phys. 178 2002 373-390
18. J. Keller R. Lewis Asymptotic methods for partial differential equations: The reduced wave equation and Maxwell's equations Surv. Appl. Math. 1 1995 1-82
19. E. Fatemi B. Engquist S.J. Osher Numerical solution of the high frequency asymptotic expansion for the scalar wave equation J. Comput. Phys. 120 1995 145-155
20. S. Gottlieb C.-W. Shu E. Tadmor Strong stability preserving high order time discretization methods SIAM Rev. 43 2000 89-112
21. J.S. Hesthaven T. Warburton Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell's equations J. Comput. Phys. 181 2002 186-221
22. G.-S. Jiang D. Peng Weighted ENO schemes for Hamilton-Jacobi equations SIAM J. Sci. Comput. 21 2000 2126-2143
23. S. Jin H.L. Liu S. Osher R. Tsai Computing multivalued physical observables for the semiclassical limit of the Schrodinger equations J. Comp. Phys. 205 1 2005 222-241
24. S. Operto S. Xu G. Lambare Can we image quantitatively complex models with rays? Geophysics 65 2000 1223-1238
25. S. Osher L.-T. Cheng M. Kang H. Shim Y.-H. Tsai Geometric optics in a phase-space-based level set and Eulerian framework J. Comput. Phys. 179 2002 622-648
26. D. Peng B. Merriman S. Osher H. Zhao M. Kang A PDE-based fast local level set method J. Comput. Phys. 155 1999 410-438
27. J. Qian L.-T. Cheng S.J. Osher A level set based Eulerian approach for anisotropic wave propagations Wave Motion 37 2003 365-379
28. J. Qian S. Leung A level set method for paraxial multivalued traveltimes J. Comput. Phys. 197 2004 711-736
29. C.W. Shu S. Osher Efficient implementation of essentially nonoscillatory shock-capturing schemes J. Comput. Phys. 77 1988 439-471
30. J.M. Song C.-C. Lu W.C. Chew Multilevel fast multiple algorithm for electromagnetic scattering by large complex objects IEEE Trans. Antennas Propagat. 45 1997 1488-1493
31. R.J. Spiteri S.J. Ruuth A new class of optimal high-order strong-stability-preserving time discretization methods SIAM J. Numer. Anal. 40 2002 469-491
32. M. Sussman M.Y. Hussaini A discontinuous spectral element method for the level set equation J. Sci. Comput. 19 2003 479-500
33. W.W. Symes, A slowness matching finite difference method for traveltimes beyond transmission caustics, in: Proceedings of the 68th Annual International Meeting, Soc. Expl. Geophys., Expanded Abstracts, 1998, pp. 1945-1948
34. W.W. Symes J. Qian A slowness matching Eulerian method for multivalued solutions of eikonal equations J. Sci. Comp. 19 2003 501-526
35. J. van Trier W.W. Symes Upwind finite-difference calculation of traveltimes Geophysics 56 1991 812-821
36. J. Vidale Finite-difference calculation of traveltimes Bull. Seism. Soc. Am. 78 1988 2062-2076
37. M.D. White, M.R. Visbal, Implicit high-order generalized coordinate solution of Maxwell's equations, in: Antennas and Propagation Society International Symposium, AP-S. Digest, vol. 3, 2002, pp. 256-259
38. G.B. Whitham Linear and Nonlinear Waves 1974 Wiley New York