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Value iteration convergence of $\epsilon$-monotone schemes for stationary Hamilton-Jacobi equations

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  • We present an abstract convergence result for the fixed point approximation of stationary Hamilton--Jacobi equations. The basic assumptions on the discrete operator are invariance with respect to the addition of constants, $\epsilon$-monotonicity and consistency. The result can be applied to various high-order approximation schemes which are illustrated in the paper. Several applications to Hamilton--Jacobi equations and numerical tests are presented.
    Mathematics Subject Classification: Primary: 65M12, 49L25; Secondary: 65M06, 65M08.


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  • [1]

    R. Abgrall, Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes, Comm. Pure Appl. Math., 49 (1996), 1339-1373.doi: 10.1002/(SICI)1097-0312(199612)49:12<1339::AID-CPA5>3.0.CO;2-B.


    R. Abgrall, Numerical discretization of boundary conditions for first order Hamilton-Jacobi equations, SIAM J. Numer. Anal., 41 (2003), 2233-2261.doi: 10.1137/S0036142998345980.


    S. Augoula and R. Abgrall, High order numerical discretization for Hamilton-Jacobi equations on triangular meshes, J. Sci. Comput., 15 (2000), 197-229.doi: 10.1023/A:1007633810484.


    D. S. Balsara, T. Rumpf, M. Dumbser and C.-D. Munz, Efficient, high accuracy ader-weno schemes for hydrodynamics and divergence-free magnetohydrodynamics, J. Comput. Phys., 228 (2009), 2480-2516.doi: 10.1016/j.jcp.2008.12.003.


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


    G. Barles, Solutions de Viscositè des Equations d'Hamilton-Jacobi, Springer-Verlag, 1998.


    G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283.


    F. Bauer, L. Grüne and W. Semmler, Adaptive spline interpolation for Hamilton-Jacobi-Bellman equations, Appl. Numer. Math., 56 (2006), 1196-1210.doi: 10.1016/j.apnum.2006.03.011.


    O. Bokanowski, E. Cristiani and H. Zidani, An efficient data structure and accurate scheme to solve front propagation problems, J. Sci. Comput., 42 (2010), 251-273.doi: 10.1007/s10915-009-9329-6.


    O. Bokanowski, J. Garcke, M. Griebel and I. Klompmaker, An adaptive sparse grid semi-Lagrangian scheme for first order Hamilton-Jacobi Bellman equations, J. Sci. Comput., 55 (2013), 575-605.doi: 10.1007/s10915-012-9648-x.


    O. Bokanowski, N. Megdich and H. Zidani, Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous initial data, Numer. Math., 115 (2010), 1-44.doi: 10.1007/s00211-009-0271-1.


    O. Bokanowski and H. Zidani, Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellmann equations, J. Sci. Comput., 30 (2007), 1-33.doi: 10.1007/s10915-005-9017-0.


    S. Bryson, A. Kurganov, D. Levy and G. Petrova, Semi-discrete central-upwind schemes with reduced dissipation for Hamilton-Jacobi equations, IMA J. Numer. Anal., 25 (2005), 113-138.doi: 10.1093/imanum/drh015.


    S. Bryson and D. Levy, High-order central WENO schemes for multidimensional Hamilton-Jacobi equations, SIAM J. Numer. Anal., 41 (2003), 1339-1369 (electronic).doi: 10.1137/S0036142902408404.


    S. Bryson and D. Levy, Mapped WENO and weighted power ENO reconstructions in semi-discrete central schemes for Hamilton-Jacobi equations, Appl. Numer. Math., 56 (2006), 1211-1224.doi: 10.1016/j.apnum.2006.03.005.


    F. Camilli, L. Grüne and F. Wirth, A generalization of Zubov's method to perturbed system, SIAM J. Control Optim., 40 (2001), 496-515.doi: 10.1137/S036301299936316X.


    I. Capuzzo Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Appl. Math. Optim., 11 (1984), 161-181.doi: 10.1007/BF01442176.


    E. Carlini, M. Falcone, and R. Ferretti., An efficient algorithm for Hamilton-Jacobi equations in high dimension, Comput. Vis. Sci., 7 (2004), 15-29.doi: 10.1007/s00791-004-0124-5.


    E. Carlini, R. Ferretti and G. Russo, A weighted essentially nonoscillatory, large time-step scheme for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 27 (2005), 1071-1091.doi: 10.1137/040608787.


    F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp., 57 (1991), 169-210.doi: 10.1090/S0025-5718-1991-1079010-2.


    F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: A general theory, SIAM J. Numer. Anal., 30 (1993), 675-700.doi: 10.1137/0730033.


    L. Corrias, M. Falcone and R. Natalini, Numerical schemes for conservation laws via Hamilton-Jacobi equations, Math. Comp., 64 (1995), 555-580, S13-S18.doi: 10.1090/S0025-5718-1995-1265013-5.


    M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp., 43 (1984), 1-19.doi: 10.1090/S0025-5718-1984-0744921-8.


    M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390.doi: 10.1090/S0002-9939-1980-0553381-X.


    M. Falcone, Numerical methods for differential games via PDEs, Int. Game Theor. Rev., 8 (2006), 231-272.doi: 10.1142/S0219198906000886.


    M. Falcone and R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations, Numer. Math., 67 (1994), 315-344.doi: 10.1007/s002110050031.


    M. Falcone and R. Ferretti, Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods, J. Comp. 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, Philadelphia, 2014.doi: 10.1137/1.9781611973051.


    R. Ferretti, Convergence of semi-Lagrangian approximations to convex Hamilton-Jacobi equations under (very) large Courant numbers, SIAM J. Numer. Anal., 40 (2002), 2240-2253.doi: 10.1137/S0036142901388378.


    B. D. Froese and A. M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal., 51 (2013), 423-444.doi: 10.1137/120875065.


    E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, volume 118 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996.doi: 10.1007/978-1-4612-0713-9.


    L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337.doi: 10.1007/s002110050241.


    L. Grüne, M. Kato and W. Semmler, Solving ecological management problems using dynamic programming, J. Econ. Behav. Organ., 57 (2005), 448-473.doi: 10.1016/j.jebo.2005.04.002.


    A. Harten, B. Engquist, S. Osher and S. R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes III, J. Comput. Phys., 71 (1987), 231-303.doi: 10.1016/0021-9991(87)90031-3.


    A. Harten and S. Osher, Uniformly high-order accurate nonoscillatory schemes. I, SIAM J. Numer. Anal., 24 (1987), 279-309.doi: 10.1137/0724022.


    A. Harten, S. Osher, B. Engquist and S. R. Chakravarthy, Some results on uniformly high-order accurate essentially nonoscillatory schemes, Appl. Numer. Math., 2 (1986), 347-377.doi: 10.1016/0168-9274(86)90039-5.


    P. Hoch and O. Pironneau, A vector Hamilton-Jacobi formulation for the numerical simulation of Euler flows, C. R. Math. Acad. Sci. Paris, 342 (2006), 151-156.doi: 10.1016/j.crma.2005.11.007.


    C. Hu and C.-W. Shu, A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (1999), 666-690.doi: 10.1137/S1064827598337282.


    G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202-228.doi: 10.1006/jcph.1996.0130.


    F. Li and C.-W. Shu, Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations, Appl. Math. Lett., 18 (2005), 1204-1209.doi: 10.1016/j.aml.2004.10.009.


    P. Lions and P. Souganidis, Convergence of MUSCL and filtered schemes for scalar conservation laws and Hamilton-Jacobi equations, Num. Math., 69 (1995), 441-470.doi: 10.1007/s002110050102.


    X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200-212.doi: 10.1006/jcph.1994.1187.


    S. Osher, Convergence of generalized MUSCL schemes, SIAM J. Numer. Anal., 22 (1985), 947-961.doi: 10.1137/0722057.


    B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys., 135 (1997), 227-248.doi: 10.1006/jcph.1997.5757.


    Y.-T. Zhang and C.-W. Shu, High-order WENO schemes for Hamilton-Jacobi equations on triangular meshes, SIAM J. Sci. Comput., 24 (2002), 1005-1030.doi: 10.1137/S1064827501396798.

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