May  2014, 19(3): 629-650. doi: 10.3934/dcdsb.2014.19.629

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

1. 

Université Paris Diderot, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, Sorbonne Paris Cité F-75205 Paris, France

2. 

"Sapienza" Università di Roma, Dip. di Scienze di Base e Applicate per l'Ingegneria, via Scarpa 16, 0161 Roma, Italy

3. 

Laboratoire d'Analyse et Probabilité, Université d'Evry Val d'Essonne, 23 Bd. de France, F-91037 Evry Cedex

Received  January 2013 Revised  October 2013 Published  February 2014

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.
Citation: Yves Achdou, Fabio Camilli, Lucilla Corrias. On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 629-650. doi: 10.3934/dcdsb.2014.19.629
References:
[1]

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

[2]

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.  doi: 10.1081/PDE-120021187.  Google Scholar

[3]

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

[4]

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

[5]

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.   Google Scholar

[6]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,, Progress in Nonlinear Differential Equations and their Applications, (2004).   Google Scholar

[7]

, P. Cardaliaguet,, Notes on Mean Fiels games (from P.-L. Lions' lectures at the Collège de France)., ().   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations,, SIAM, (2013).  doi: 10.1137/1.9781611973051.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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.  doi: 10.1080/03605309708821265.  Google Scholar

[27]

R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970).   Google Scholar

[28]

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

[29]

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

show all references

References:
[1]

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

[2]

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.  doi: 10.1081/PDE-120021187.  Google Scholar

[3]

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

[4]

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

[5]

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.   Google Scholar

[6]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,, Progress in Nonlinear Differential Equations and their Applications, (2004).   Google Scholar

[7]

, P. Cardaliaguet,, Notes on Mean Fiels games (from P.-L. Lions' lectures at the Collège de France)., ().   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations,, SIAM, (2013).  doi: 10.1137/1.9781611973051.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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.  doi: 10.1080/03605309708821265.  Google Scholar

[27]

R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1970).   Google Scholar

[28]

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

[29]

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

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