Geodesic
The plane wave refraction on convex and concave obtuse angles in geometric acoustics approximation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 3, pp. 251-271.

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The strict analytical solution to the eikonal equation for the plane wave refracted on convex and concave obtuse angles has been built. It has a shock line for the ray vector field and the first arrival times at the convex angle and a rarefaction cone with diffracted waves at the concave angle. This cone corresponds to the Keller diffraction cone in the geometric diffraction theory. The comparison of the first arrival times, the Hamilton-Jacoby equation times for downward waves and the conservation ray parameter equation times was made. It is shown that these times are equal only for pre-critical incident angles and are different for sub-critical angles. It is shown that the most energetic wave arrival times, which have dominant practical importance, are equal to the times calculated for the conservation ray parameter equation. The numerical algorithm proposed for these times calculation may be used for arbitrary velocity models. 
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A. N. Kremlev. The plane wave refraction on convex and concave obtuse angles in geometric acoustics approximation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 3, pp. 251-271. http://geodesic.mathdoc.fr/item/SJVM_2017_20_3_a2/

[1] Alekseev A. S., Babich V. M., Gelchinskii B. Ya., “Luchevoi metod vychisleniya intensivnosti volnovykh frontov”, Voprosy dinamicheskoi teorii rasprostraneniya seismicheskikh voln, 5 (1961), 3–24

[2] Cerveny V., Seismic Ray Theory, Cambridge University Press, Cambridge, 2001 | MR | Zbl

[3] Uizem D., Lineinye i nelineinye volny, Per. s angl. pod red. A. B. Shabata, Mir, M., 1977

[4] Aki K., Richards P., Kolichestvennaya seismologiya. Teoriya i metody, v. 1, Mir, M., 1983

[5] Sethian J. A., “A fast marching level set method for monotonically advancing fronts”, Proc. of the National Academy of Sciences, 93:4 (1996), 1591–1595 | DOI | MR | Zbl

[6] Zhao H., “A fast sweeping method for eikonal equations”, Mathematics of Computation, 74 (2005), 603–627 | DOI | MR | Zbl

[7] Vidale J., “Finite-difference calculation of travel times”, Bull. of the Seismological Society of America, 78:6 (1988), 2062–2076

[8] Podvin P., Lecomte I., “Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools”, Geophysical J. Int., 105:1 (1991), 271–284 | DOI

[9] Crandall M., Lions P., “Viscosity solutions of Hamilton–Jacobi equations”, Trans. Amer. Math. Soc., 277 (1983), 1–42 | DOI | MR | Zbl

[10] Nikitin A. A., Serdyukov A. S., Duchkov A. A., “Parallelnyi algoritm resheniya uravneniya eikonala dlya trekhmernykh zadach seismorazvedki”, Vestnik NGU. Seriya: Informatsionnye tekhnologii, 13:3 (2015), 19–28

[11] Kim S., “The most-energetic traveltime of seismic waves”, Applied Mathematics Letters, 14 (2001), 313–319 | DOI | Zbl

[12] Geoltrain S., Brac J., “Can we image complex structures with first-arrival traveltime”, Geophysics, 58:4 (1993), 564–575 | DOI

[13] Van Trier J., Symes W. W., “Upwind finite-difference calculation of traveltimes”, Geophysics, 56 (1991), 812–821 | DOI

[14] Belfi C. D., Second and Third Order ENO Methods for the Eikonal Equation., Technical report/ The Rice Inversion Project, Rice University, Houston, Texas, USA, 1997

[15] Quin J., Symes W. W., “An adaptive finite-difference method for traveltimes and amplitudes”, Geophysics, 67 (2002), 167–176 | DOI

[16] Karlsen K. H., Towers J. D., “Convergence of the Lax–Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux”, Chinese Ann. Math. Ser. B, 25 (2004), 287–318 | DOI | MR | Zbl

[17] LeVeque R. J., Numerical Methods for Conservation Laws, Lectures in Mathematics, ETH Zurich, Birkhauser Verlag, Basel, 1990 | MR | Zbl

[18] Keller J. B., “Geometrical theory of diffraction”, J. Opt. Soc. Am., 52 (1962), 116–130 | DOI | MR

[19] Klem-Musatov K. D., Aizenberg A. M., “Seismic modeling by methods of the theory of edge waves”, Geophysics, 57 (1985), 90–105

[20] Godunov S. K., “Raznostnyi metod chislennogo rascheta razryvnykh reshenii uravnenii gidrodinamiki”, Matematicheskii sbornik, 47(89):3 (1959), 271–306 | MR | Zbl

[21] Jeong W.-K., Whitaker R., “A fast iterative method for eikonal equations”, SIAM J. on Scientific Computing, 30:5 (2008), 2512–2534 | DOI | MR | Zbl

[22] Crandall M. G., Evans L. C., Lions P. L., “Some properties of viscosity solutions of Hamilton–Jacobi equations”, Trans. of the American Mathematical Society, 282:2 (1984), 487–502 | DOI | MR | Zbl