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On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations

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  • In the present article, we study the numerical approximation of a system of Hamilton-Jacobi and transport equations arising in geometrical optics. We consider a semi-Lagrangian scheme. We prove the well posedness of the discrete problem and the convergence of the approximated solution toward the viscosity-measure valued solution of the exact problem.
    Mathematics Subject Classification: Primary: 65M12; Secondary: 35F25, 35R05, 49L25.


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  • [1]

    M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston Inc., Boston, MA, 1997.doi: 10.1007/978-0-8176-4755-1.


    B. Ben Moussa and G. T. Kossioris, On the system of Hamilton-Jacobi and transport equations arising in geometrical optics, Comm. Partial Differential Equations, 28 (2003), 1085-1111.doi: 10.1081/PDE-120021187.


    H. A. Bethe and E. E. Salpeter, A Relativistic Equation for Bound-State Problems, Phys. Rev., 84 (1951), 1232-1242.doi: 10.1103/PhysRev.84.1232.


    F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis, TMA, 32 (1998), 891-933.doi: 10.1016/S0362-546X(97)00536-1.


    Y. Brenier, Un algorithme rapide pour le calcul de transformée de Legendre-Fenchel discrètes, C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 587-589.


    P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, MA, 2004.


    R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.doi: 10.1142/9789812793133.


    L. Corrias, Fast Legendre-Fenchel Transform and Applications to Hamilton-Jacobi Equations and Conservation Laws, SIAM J. Num. Analysis, 33 (1996), 1534-1558.doi: 10.1137/S0036142993260208.


    L. Corrias, M. Falcone and R. Natalini, Numerical schemes for conservation laws via Hamilton-Jacobi equations, Math. Comp., 64 (1995), 555-580.doi: 10.1090/S0025-5718-1995-1265013-5.


    M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull AMS, 27 (1992), 1-67.doi: 10.1090/S0273-0979-1992-00266-5.


    M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 277 (1983), 1-42.doi: 10.1090/S0002-9947-1983-0690039-8.


    E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182.doi: 10.1137/100797515.


    M. Falcone and R. Ferretti, Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods, J. Comput. Phys., 175 (2002), 559-575.doi: 10.1006/jcph.2001.6954.


    M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, SIAM, Other Titles in Applied Mathematics, Published, 2013.doi: 10.1137/1.9781611973051.


    D. A. Gomes, Viscosity solution methods and the discrete Aubry-Mather problem, Discrete Contin. Dyn. Syst., 13, (2005), 103-116.doi: 10.3934/dcds.2005.13.103.


    L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comp., 69 (2000), 987-1015.doi: 10.1090/S0025-5718-00-01185-6.


    L. Gosse and F. James, Convergence results for an inhomogeneous system arising in various high frequency approximations, Numer. Math., 90 (2002), 721-753.doi: 10.1007/s002110100309.


    J. M. Lasry and P. L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.doi: 10.1007/s11537-007-0657-8.


    P. Lax and X. Liu, Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws II, J. Comp.Physics, 187 (2003), 428-440.doi: 10.1016/S0021-9991(03)00100-1.


    C. T. Lin and E. Tadmor, $L^1$-stability and error estimates for approximate Hamilton-Jacobi solutions, Numer. Math., 87 (2001), 701-735.doi: 10.1007/PL00005430.


    Y. Lucet, Faster than the Fast Legendre Transform, the Linear-time Legendre Transform, Numerical Algorithms, 16 (1997), 171-185.doi: 10.1023/A:1019191114493.


    Y. Lucet, What shape is your conjugate? a survey of computational convex analysis and its applications, SIGEST section of SIAM Review, 52 (2010), 505-542.doi: 10.1137/100788458.


    B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion modeling, Netw. Heterog. Media, 6 (2011), 485-519.doi: 10.3934/nhm.2011.6.485.


    G. Petrova and B. Popov, Linear transport equations with discontinuous coefficients, Comm. Partial Differential Equations, 24 (1999), 1849-1873.doi: 10.1080/03605309908821484.


    F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Comm. Partial Differential Equations, 22 (1997), 337-358.doi: 10.1080/03605309708821265.


    R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.


    P. E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. Differential Equations, 59 (1985), 1-43.doi: 10.1016/0022-0396(85)90136-6.


    T. Strömberg, Well-posedness for the system of the Hamilton-Jacobi and the continuity equations, J. Evol. Equ., 7 (2007), 669-700.doi: 10.1007/s00028-007-0327-6.

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