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

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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

Yves Achdou ^1, , Fabio Camilli ^2, and Lucilla Corrias ^3,

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

Abstract
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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.

Keywords: numerical approximations, Hamilton-Jacobi equation, transport equation, semiconcavity, OSL condition., measure solutions, viscosity solutions.

Mathematics Subject Classification: Primary: 65M12; Secondary: 35F25, 35R05, 49L2.

Citation: Yves Achdou, Fabio Camilli, Lucilla Corrias. On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations. Discrete and 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., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.
[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), 1085-1111. doi: 10.1081/PDE-120021187.
[3]	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.
[4]	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.
[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), 587-589.
[6]	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.
[7]	, P. Cardaliaguet, Notes on Mean Fiels games (from P.-L. Lions' lectures at the Collège de France).
[8]	R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812793133.
[9]	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.
[10]	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.
[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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[12]	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.
[13]	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.
[14]	M. Falcone and R. Ferretti, Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods, J. Comput. Phys., 175 (2002), 559-575. doi: 10.1006/jcph.2001.6954.
[15]	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.
[16]	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.
[17]	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.
[18]	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.
[19]	J. M. Lasry and P. L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.
[20]	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.
[21]	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.
[22]	Y. Lucet, Faster than the Fast Legendre Transform, the Linear-time Legendre Transform, Numerical Algorithms, 16 (1997), 171-185. doi: 10.1023/A:1019191114493.
[23]	Y. Lucet, What shape is your conjugate? a survey of computational convex analysis and its applications, SIGEST section of SIAM Review, 52 (2010), 505-542. doi: 10.1137/100788458.
[24]	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.
[25]	G. Petrova and B. Popov, Linear transport equations with discontinuous coefficients, Comm. Partial Differential Equations, 24 (1999), 1849-1873. doi: 10.1080/03605309908821484.
[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), 337-358. doi: 10.1080/03605309708821265.
[27]	R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.
[28]	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.
[29]	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.

show all references

References:

[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.
[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), 1085-1111. doi: 10.1081/PDE-120021187.
[3]	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.
[4]	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.
[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), 587-589.
[6]	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.
[7]	, P. Cardaliaguet, Notes on Mean Fiels games (from P.-L. Lions' lectures at the Collège de France).
[8]	R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812793133.
[9]	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.
[10]	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.
[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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[12]	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.
[13]	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.
[14]	M. Falcone and R. Ferretti, Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods, J. Comput. Phys., 175 (2002), 559-575. doi: 10.1006/jcph.2001.6954.
[15]	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.
[16]	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.
[17]	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.
[18]	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.
[19]	J. M. Lasry and P. L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.
[20]	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.
[21]	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.
[22]	Y. Lucet, Faster than the Fast Legendre Transform, the Linear-time Legendre Transform, Numerical Algorithms, 16 (1997), 171-185. doi: 10.1023/A:1019191114493.
[23]	Y. Lucet, What shape is your conjugate? a survey of computational convex analysis and its applications, SIGEST section of SIAM Review, 52 (2010), 505-542. doi: 10.1137/100788458.
[24]	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.
[25]	G. Petrova and B. Popov, Linear transport equations with discontinuous coefficients, Comm. Partial Differential Equations, 24 (1999), 1849-1873. doi: 10.1080/03605309908821484.
[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), 337-358. doi: 10.1080/03605309708821265.
[27]	R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.
[28]	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.
[29]	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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