Runborg, Olof
Some new results in multiphase geometrical optics
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 34 (2000) no. 6 , p. 1203-1231
Zbl 0972.78001 | MR 1812734
URL stable : http://www.numdam.org/item?id=M2AN_2000__34_6_1203_0

Bibliographie

[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. Zbl 0860.65052

[3] J.-D. Benamou, Direct solution of multivalued phase space solutions for Hamilton-Jacobi equations. Comm. Pure Appl. Math. 52 (1999) 1443-1475. MR 1702708 | Zbl 0935.35032

[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). Numdam | MR 1813168 | Zbl 1113.35334

[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. MR 1323183 | Zbl 0863.76068

[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. MR 1332618 | Zbl 0829.35139

[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. MR 1720754 | Zbl 0937.35098

[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. Numdam | MR 1614638 | Zbl 0893.35068

[9] Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35 (1998) 2317-2328. MR 1655848 | Zbl 0924.35080

[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). Zbl pre01786302

[11] M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math., Soc. 277 (1983) 1-42. MR 690039 | Zbl 0599.35024

[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. MR 1384139 | Zbl 0852.35097

[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. MR 1345031 | Zbl 0836.65099

[14] B. Engquist and O. Runborg, Multiphase computations in geometrical optics. J. Comput. Appl. Math. 74 (1996) 175-192. MR 1430373 | Zbl 0947.78001

[15] B. Engquist and O. Runborg, Multiphase computations in geometricai opties, in Hyperbolic Problems : Theory, Numerics, Applications, M. Fey and R. Jeltsch Eds., Internat. Ser. Numer. Math. 129, ETH Zentrum, Zurich, Switzerland (1998). Zbl 0963.78004

[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. MR 1438151 | Zbl 0881.35099

[17] L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients. Math. Comp. 69 (2000) 987-1015. MR 1670896 | Zbl 0949.65094

[18] H. Grad, On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949) 331-407. MR 33674 | Zbl 0037.13104

[19] E. Grenier, Existence globale pour le système des gaz sans pression. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 171-174. MR 1345441 | Zbl 0837.35088

[20] G.-S. Jiang and E. Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (1998) 1892-1917. MR 1638064 | Zbl 0914.65095

[21] J. Keller, Geometrical theory of diffraction. J. Opt Soc. Amer. 52 (1962) 116-130. MR 135064 | Zbl 0092.20604

[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). MR 1153252 | Zbl 0847.65053

[25] C. D. Levermore, Moment closure hierarchies for kinetic théories. J. Stat Phys. 83 (1996) 1021-1065. MR 1392419 | Zbl 1081.82619

[26] P.-L. Lions and T. Paul, Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 (1993) 553-618. MR 1251718 | Zbl 0801.35117

[27] D. Ludwig, Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19 (1966) 215-250. MR 196254 | Zbl 0163.13703

[28] S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907-922. MR 1111446 | Zbl 0736.65066

[29] F. Poupaud and C.-W. Shu, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients. Comm. Partial Differential Equations 22 (1997) 337-358. MR 1434148 | Zbl 0882.35026

[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. MR 1069518 | Zbl 0774.35008

[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). MR 483954 | Zbl 0373.76001

[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. MR 1690841 | Zbl 0929.35089