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.
@article{M2AN_2002__36_5_883_0,
author = {Benamou, Jean-David and Hoch, Philippe},
title = {GO++ : a modular lagrangian/eulerian software for Hamilton Jacobi equations of geometric optics type},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {36},
year = {2002},
pages = {883-905},
doi = {10.1051/m2an:2002037},
mrnumber = {1955540},
zbl = {1023.78001},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2002__36_5_883_0}
}
Benamou, Jean-David; Hoch, Philippe. GO++ : a modular lagrangian/eulerian software for Hamilton Jacobi equations of geometric optics type. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) pp. 883-905. doi : 10.1051/m2an:2002037. http://gdmltest.u-ga.fr/item/M2AN_2002__36_5_883_0/
[1] and, 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] , Mathematical methods of Classical Mechanics. Springer-Verlag (1978). | Zbl 0386.70001
[3] , Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag (1994). | Zbl 0819.35002
[4] , Big ray tracing: Multi-valued travel time field computation using viscosity solutions of the eikonal equation. J. Comput. Phys. 128 (1996) 463-474. | Zbl 0860.65052
[5] , Direct solution of multi-valued phase-space solutions for Hamilton-Jacobi equations. Comm. Pure Appl. Math. 52 (1999). | Zbl 0935.35032
[6] and, GO++: A modular Lagrangian/Eulerian software for Hamilton-Jacobi equations of Geometric Optics type. INRIA Tech. Report RR.
[7] and, 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 0893.35068
[8] and, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42. | Zbl 0599.35024
[9] , Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27 (1974) 207-281. | Zbl 0285.35010
[10] , and, Numerical resolution of the high frequency asymptotic expansion of the scalar wave equation. J. Comput. Phys. 120 (1995) 145-155. | Zbl 0836.65099
[11] and, Multi-phase computation in geometrical optics. Tech report, Nada KTH (1995). | MR 1430373 | Zbl 0947.78001
[12] , The theory of Legendrian unfoldings and first order differential equations. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 517-532. | Zbl 0786.35033
[13] , and, Two dimensional multi-valued traveltime and amplitude maps by uniform sampling of a ray field. Geophys. J. Int 125 (1996) 584-598.
[14] S. Ruuth and S.J. Osher, A fixed grid method for capturing the motion of self-intersecting interfaces and related PDEs. Preprint (1999).
[15] and, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 83 (1989) 32-78. | Zbl 0674.65061
[16] , and, A new eulerian method for the computation of propagating short acoustic and electromagnetic pulses. J. Comput. Phys. 157 (2000) 683-706. | Zbl 1043.78556
[17] , A slowness matching algorithm for multiple traveltimes. TRIP report (1996).
[18] , and, Traveltime and amplitude estimation using wavefront construction. Geophysics 58 (1993) 1157-1166.
[19] , Lecture on the Calculus of Variation and Optimal Control Theory. Saunders, Philadelphia (1969). | MR 259704 | Zbl 0177.37801