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September  2015, 35(9): 4041-4070. doi: 10.3934/dcds.2015.35.4041

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

Olivier Bokanowski 1, , Maurizio Falcone 2, , Roberto Ferretti 3, , Lars Grüne 4, , Dante Kalise 5, and  Hasnaa Zidani 6,

1. 

Laboratoire Jacques-Louis Lions, UMR 7598, Université Paris-Diderot (Paris 7), UFR de Mathématiques - 5 rue Thomas Mann, 75205 Paris CEDEX 13
2. 

Dipartimento di Matematica, Istituto "Guido Castelnuovo", Sapienza Università di Roma, Piazzale Aldo Moro, 2 I-00185 Roma
3. 

Dipartimento di Matematica e Fisica, Università di Roma Tre, L.go S. Leonardo Murialdo, 1, 00146 Roma, Italy
4. 

Mathematisches Institut, Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, 95440 Bayreuth, Germany
5. 

Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria
6. 

Unité des mathématiques appliquées (UMA), ENSTA ParisTech, 828 Bd Maréchaux, 91120 Palaiseau

Received  April 2014 Published  April 2015

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.
Keywords: $\epsilon$-monotonicity, high-order methods., fixed point approximation schemes, Hamilton--Jacobi equation.
Mathematics Subject Classification: Primary: 65M12, 49L25; Secondary: 65M06, 65M0.
Citation: Olivier Bokanowski, Maurizio Falcone, Roberto Ferretti, Lars Grüne, Dante Kalise, Hasnaa Zidani. Value iteration convergence of $\epsilon$-monotone schemes for stationary Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4041-4070. doi: 10.3934/dcds.2015.35.4041
References:
[1]

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

[2]

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

[3]

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

[4]

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. doi: 10.1016/j.jcp.2008.12.003. Google Scholar

[5]

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. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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. doi: 10.1007/s10915-009-9329-6. Google Scholar

[10]

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. doi: 10.1007/s10915-012-9648-x. Google Scholar

[11]

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. doi: 10.1007/s00211-009-0271-1. Google Scholar

[12]

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

[13]

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. doi: 10.1093/imanum/drh015. Google Scholar

[14]

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

[15]

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. doi: 10.1016/j.apnum.2006.03.005. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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. doi: 10.1137/040608787. Google Scholar

[20]

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. doi: 10.1090/S0025-5718-1991-1079010-2. Google Scholar

[21]

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. doi: 10.1137/0730033. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations,, SIAM, (2014). doi: 10.1137/1.9781611973051. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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. doi: 10.1016/0021-9991(87)90031-3. Google Scholar

[35]

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

[36]

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. doi: 10.1016/0168-9274(86)90039-5. Google Scholar

[37]

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. doi: 10.1016/j.crma.2005.11.007. Google Scholar

[38]

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

[39]

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

[40]

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. doi: 10.1016/j.aml.2004.10.009. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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. doi: 10.1137/S1064827501396798. Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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. doi: 10.1016/j.jcp.2008.12.003. Google Scholar

[5]

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. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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. doi: 10.1007/s10915-009-9329-6. Google Scholar

[10]

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. doi: 10.1007/s10915-012-9648-x. Google Scholar

[11]

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. doi: 10.1007/s00211-009-0271-1. Google Scholar

[12]

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

[13]

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. doi: 10.1093/imanum/drh015. Google Scholar

[14]

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

[15]

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. doi: 10.1016/j.apnum.2006.03.005. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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. doi: 10.1137/040608787. Google Scholar

[20]

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. doi: 10.1090/S0025-5718-1991-1079010-2. Google Scholar

[21]

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. doi: 10.1137/0730033. Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations,, SIAM, (2014). doi: 10.1137/1.9781611973051. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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. doi: 10.1016/0021-9991(87)90031-3. Google Scholar

[35]

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

[36]

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. doi: 10.1016/0168-9274(86)90039-5. Google Scholar

[37]

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. doi: 10.1016/j.crma.2005.11.007. Google Scholar

[38]

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

[39]

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

[40]

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. doi: 10.1016/j.aml.2004.10.009. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[45]

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. doi: 10.1137/S1064827501396798. Google Scholar

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