GO++ : a modular lagrangian/eulerian software for Hamilton Jacobi equations of geometric optics type
ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 5, pp. 883-905.

We describe both the classical lagrangian and the Eulerian methods for first order Hamilton-Jacobi equations of geometric optic type. We then explain the basic structure of the software and how new solvers/models can be added to it. A selection of numerical examples are presented.

DOI : 10.1051/m2an:2002037
Classification : 78A05, 78H20
Mots clés : Hamilton-Jacobi, hamiltonian system, ray tracing, viscosity solution, upwind scheme, geometric optics, C++
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     title = {GO++ : a modular lagrangian/eulerian software for {Hamilton} {Jacobi} equations of geometric optics type},
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Benamou, Jean-David; Hoch, Philippe. GO++ : a modular lagrangian/eulerian software for Hamilton Jacobi equations of geometric optics type. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 5, pp. 883-905. doi : 10.1051/m2an:2002037. http://archive.numdam.org/articles/10.1051/m2an:2002037/

[1] R. Abgrall and J.-D. Benamou, Big ray tracing and eikonal solver on unstructured grids: Application to the computation of a multi-valued travel-time field in the marmousi model. Geophysics 64 (1999) 230-239.

[2] V.I. Arnol'D, Mathematical methods of Classical Mechanics. Springer-Verlag (1978). | Zbl

[3] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag (1994). | Zbl

[4] J.-D. Benamou, Big ray tracing: Multi-valued travel time field computation using viscosity solutions of the eikonal equation. J. Comput. Phys. 128 (1996) 463-474. | Zbl

[5] J.-D. Benamou, Direct solution of multi-valued phase-space solutions for Hamilton-Jacobi equations. Comm. Pure Appl. Math. 52 (1999). | Zbl

[6] J.-D. Benamou and P. Hoch, GO++: A modular Lagrangian/Eulerian software for Hamilton-Jacobi equations of Geometric Optics type. INRIA Tech. Report RR.

[7] Y. Brenier and L. Corrias, A kinetic formulation for multi-branch entropy solutions of scalar conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 169-190. | Numdam | Zbl

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

[9] J.J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27 (1974) 207-281. | Zbl

[10] B. Engquist, E. Fatemi and S. Osher, Numerical resolution of the high frequency asymptotic expansion of the scalar wave equation. J. Comput. Phys. 120 (1995) 145-155. | Zbl

[11] B. Engquist and O. Runborg, Multi-phase computation in geometrical optics. Tech report, Nada KTH (1995). | MR | Zbl

[12] S. Izumiya, The theory of Legendrian unfoldings and first order differential equations. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 517-532. | Zbl

[13] G. Lambare, P. Lucio and A. Hanyga, Two dimensional multi-valued traveltime and amplitude maps by uniform sampling of a ray field. Geophys. J. Int 125 (1996) 584-598.

[14] B. Merryman S. Ruuth and S.J. Osher, A fixed grid method for capturing the motion of self-intersecting interfaces and related PDEs. Preprint (1999).

[15] S.J. Osher and C.W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 83 (1989) 32-78. | Zbl

[16] J. Steinhoff, M. Fang and L. Wang, A new eulerian method for the computation of propagating short acoustic and electromagnetic pulses. J. Comput. Phys. 157 (2000) 683-706. | Zbl

[17] W. Symes, A slowness matching algorithm for multiple traveltimes. TRIP report (1996).

[18] V. Vinje, E. Iversen and H. Gjoystdal, Traveltime and amplitude estimation using wavefront construction. Geophysics 58 (1993) 1157-1166.

[19] L.C. Young, Lecture on the Calculus of Variation and Optimal Control Theory. Saunders, Philadelphia (1969). | MR | Zbl

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