GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type

Jean-David Benamou; Philippe Hoch

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 5, page 883-905
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Benamou, Jean-David, and Hoch, Philippe. "GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type." ESAIM: Mathematical Modelling and Numerical Analysis 36.5 (2010): 883-905. <http://eudml.org/doc/194131>.

@article{Benamou2010,
abstract = { 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. },
author = {Benamou, Jean-David, Hoch, Philippe},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Hamilton–Jacobi; Hamiltonian system; ray tracing; viscosity solution; upwind scheme; geometric optics; C++.; Hamilton-Jacobi equations; ray tracing; C++},
language = {eng},
month = {3},
number = {5},
pages = {883-905},
publisher = {EDP Sciences},
title = {GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type},
url = {http://eudml.org/doc/194131},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Benamou, Jean-David
AU - Hoch, Philippe
TI - GO++: A modular Lagrangian/Eulerian software for Hamilton Jacobi equations of geometric optics type
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 5
SP - 883
EP - 905
AB - 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.
LA - eng
KW - Hamilton–Jacobi; Hamiltonian system; ray tracing; viscosity solution; upwind scheme; geometric optics; C++.; Hamilton-Jacobi equations; ray tracing; C++
UR - http://eudml.org/doc/194131
ER -

References

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  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. Geophysics64 (1999) 230-239.  
  2. V.I. Arnol'd, Mathematical methods of Classical Mechanics. Springer-Verlag (1978).  
  3. G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag (1994).  
  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.  
  5. J.-D. Benamou, Direct solution of multi-valued phase-space solutions for Hamilton-Jacobi equations. Comm. Pure Appl. Math. 52 (1999).  
  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 Linéaire15 (1998) 169-190.  
  8. M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277 (1983) 1-42.  
  9. J.J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math.27 (1974) 207-281.  
  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.  
  11. B. Engquist and O. Runborg, Multi-phase computation in geometrical optics. Tech report, Nada KTH (1995).  
  12. S. Izumiya, The theory of Legendrian unfoldings and first order differential equations. Proc. Roy. Soc. Edinburgh Sect. A123 (1993) 517-532.  
  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. Int125 (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.  
  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.  
  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. Geophysics58 (1993) 1157-1166.  
  19. L.C. Young, Lecture on the Calculus of Variation and Optimal Control Theory. Saunders, Philadelphia (1969).  

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