Some new results in multiphase geometrical optics

Mathematical Modelling and Numerical Analysis

Volumn 34, Issue 6, 2000, Pages 1203-1231

  Runborg, Olof ^a  

^a PRINCETON UNIVERSITY   (United States)

Author keywords

Conservation laws; Eikonal equation; Finite difference methods; Geometrical optics; Kinetic equations; Moment equations; Multivalued traveltimes; Nonstrictly hyperbolic system

Indexed keywords

EID: 0034551460     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2000124     Document Type: Article

References (36)

1. R. Abgrall and 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. J.-D. Benamou, Big ray tracing: Multivalued travel time field computation using viscosity solutions of the eikonal equation. J. Comput. Phys. 128 (1996) 463-474.
3. J.-D. Benamou, Direct solution of multivalued phase space solutions for Hamilton-Jacobi equations. Comm. Pure Appl. Math. 52 (1999) 1443-1475.
4. J.-D. Benamou, F. Castella, T. Katsaounis and B. Perthame, High frequency limit of the Helmholtz equation. Research report DMA-99-25. Département de Mathématiques et Applications, École Normale Supérieure, Paris (1999).
5. F. Bouchut, On zero pressure gas dynamics, in Advances in kinetic theory and computing. Ser. Adv. Math. Appl. Sci. 22, World Sci. Publishing, River Edge, NJ (1994) 171-190.
6. F. Bouchut and F. James, Équations de transport unidimensionnelles à coefficients discontinus. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995) 1097-1102.
7. F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness. Comm. Partial Differential Equations 24 (1999) 2173-2189.
8. Y. Brenier and L. Corrias, A kinetic formulation for multibranch entropy solutions of scalar conservation laws. Ann. Inst. H. Poincaré 15 (1998) 169-190.
9. Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35 (1998) 2317-2328.
10. F. Castella, O. Runborg and B. Perthame, High frequency limit of the Helmholtz equation II: Source on a general smooth manifold. Research report, Département de Mathématiques et Applications, École Normale Supérieure, Paris (2000).
11. M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42.
12. W. E, Yu.G. Rykov and Ya.G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys. 177 (1996) 349-380.
13. B. Engquist, E. Fatemi and S. Osher, Numerical solution of the high frequency asymptotic expansion for the scalar wave equation. J. Comput. Phys. 120 (1995) 145-155.
14. B. Engquist and O. Runborg, Multiphase computations in geometrical optics. J. Comput. Appl. Math. 74 (1996) 175-192.
15. B. Engquist and O. Runborg, Multiphase computations in geometrical optics, in Hyperbolic Problems: Theory, Numerics, Applications, M. Fey and R. Jeltsch Eds., Internat. Ser. Numer. Math. 129, ETH Zentrum, Zürich, Switzerland (1998).
16. P. Gérard, P.A. Markowich, N.J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323-379.
17. L. Gosse and F. James. Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients. Math. Comp. 69 (2000) 987-1015.
18. H. Grad, On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331-407.
19. E. Grenier, Existence globale pour le système des gaz sans pression. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 171-174.
20. G.-S. Jiang and E. Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (1998) 1892-1917.
21. J. Keller, Geometrical theory of diffraction. J. Opt. Soc. Amer. 52 (1962) 116-130.
22. R.G. Kouyoumjian and P.H. Pathak, A uniform theory of diffraction for an edge in a perfectly conducting surface. Proc. IEEE 62 (1974) 1448-1461.
23. Yu.A. Kravtsov, On a modification of the geometrical optics method. Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7 (1964) 664-673.
24. R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser (1992).
25. C.D. Levermore, Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83 (1996) 1021-1065.
26. P.-L. Lions and T. Paul, Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 (1993) 553-618.
27. D. Ludwig, Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19 (1966) 215-250.
28. S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jaccbi equations. SIAM J. Numer. Anal. 28 (1991) 907-922.
29. F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients. Comm. Partial Differential Equations 22 (1997) 337-358.
30. O. Runborg, Multiscale and Multiphase Methods for Wave Propagation. Ph.D. thesis, Department of Numerical Analysis and Computing Science, KTH, Stockholm (1998).
31. W.W. Symes, A slowness matching finite difference method for traveltimes beyond transmission caustics. Preprint, Dept. of Computational and Applied Mathematics, Rice University (1996).
32. L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 115 (1990) 193-230.
33. J. van Trier and W.W. Symes, Upwind finite-difference calculation of traveltimes. Geophysics 56 (1991) 812-821.
34. J. Vidale, Finite-difference calculation of traveltimes. Bull. Seismol Soc. Amer. 78 (1988) 2062-2076.
35. G.B. Whitham, Linear and Nonlinear Waves. John Wiley & Sons (1974).
36. Y. Zheng, Systems of conservation laws with incomplete sets of eigenvectors everywhere, in Advances in Nonlinear Partial Differential Equations and Related Areas, World Sci. Publishing, River Edge, NJ (1998) 399-426.