A narrow-stencil finite difference method for approximating viscosity solutions of Hamilton-Jacobi-Bellman equations. (English) Zbl 1468.65174
Summary: This paper presents a new narrow-stencil finite difference method for approximating viscosity solutions of Hamilton-Jacobi-Bellman equations. The proposed finite difference scheme naturally extends the Lax-Friedrichs scheme for first order fully nonlinear PDEs to second order fully nonlinear PDEs which are approximated by Lax-Friedrichs-like numerical operators. The crux for constructing such a numerical operator is to introduce a stabilization term, which is called a “numerical moment” and corresponds to the numerical viscosity term in the original Lax-Friedrichs scheme for first order PDEs. It is proved that the proposed Lax-Friedrichs-like scheme has a unique solution and is stable in both the \(\ell^2\)-norm and the \(\ell^\infty\)-norm. Moreover, the convergence of the proposed finite difference scheme to the viscosity solution of the underlying Hamilton-Jacobi-Bellman equation is also established using a novel discrete comparison argument.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35F21 | Hamilton-Jacobi equations |
Keywords:
Hamilton-Jacobi-Bellman equations; finite difference methods; numerical operators; fully nonlinear PDEs; viscosity solutions; generalized monotonicity; numerical momentReferences:
| [1] | G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptot. Anal., 4 (1991), pp. 271-283. · Zbl 0729.65077 |
| [2] | J. F. Bonnans and H. Zidani, Consistency of generalized finite difference schemes for the stochastic HJB equation, SIAM J. Numer. Anal., 41 (2003), pp. 1008-1021. · Zbl 1130.49307 |
| [3] | L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, American Mathematical Society, Providence, RI, 1995. · Zbl 0834.35002 |
| [4] | M. G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1-67. · Zbl 0755.35015 |
| [5] | M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), pp. 1-42. · Zbl 0599.35024 |
| [6] | M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), pp. 487-502. · Zbl 0543.35011 |
| [7] | K. Debrabant and E. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Math. Comp., 82 (2013), pp.1433-1462. · Zbl 1276.65050 |
| [8] | X. Feng and M. Jensen, Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids, SIAM J. Numer. Anal., 55 (2017), pp. 691-712. · Zbl 1362.65117 |
| [9] | X. Feng and M. Neilan, Mixed finite element methods for the fully nonlinear Monge-Ampére equation based on the vanishing moment method, SIAM J. Numer. Anal. 47 (2009), pp. 1226-1250. · Zbl 1195.65170 |
| [10] | X. Feng, R. Glowinski, and M. Neilan, Recent developments in numerical methods for fully nonlinear second order partial differential equations, SIAM Rev., 55 (2013), pp. 205-267. · Zbl 1270.65062 |
| [11] | X. Feng and M. Neilan, The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis, preprint, https://arxiv.org/abs/1109.1183v2 (2016). |
| [12] | X. Feng, C. Kao, and T. Lewis, Convergent FD methods for one-dimensional fully nonlinear second order partial differential equations, J. Comput. Appl. Math., 254 (2013), pp. 81-98. · Zbl 1290.65096 |
| [13] | X. Feng and T. Lewis, A Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Elliptic Partial Differential Equations with Applications to Hamilton-Jacobi-Bellman Equations, preprint, https://arxiv.org/abs/1907.10204 (2019). |
| [14] | W. H. Fleming and H. M. Soner, Controlled Markov Process and Viscosity Solutions, Springer, New York, 2006. · Zbl 1105.60005 |
| [15] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. · Zbl 1042.35002 |
| [16] | E. Habil, Double sequences and double series, IUG Journal of Natural Studies, 14 (2006), pp. 1-32. |
| [17] | M. Jensen and I. Smears, On the convergence of finite element methods for Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal. 51 (2013), pp. 137-162. · Zbl 1266.65166 |
| [18] | H. Kushner and P. G. Dupuis, Numerical Methods for Stochastic Optimal Control Problems in Continuous Time, Appl. Math. 24, Springer, New York, 1992. · Zbl 0754.65068 |
| [19] | T. Lewis, Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations, Ph.D. Thesis, University of Tennessee, Knoxville, TN, 2013. |
| [20] | T. S. Motzkin and W. Wasow, On the approximation of linear elliptic differential equations by difference equations with positive coefficients, J. Math. Phys., 31 (1953), pp. 253-259. · Zbl 0050.12501 |
| [21] | M. Neilan, A.J. Salgado and W. Zhang, Numerical analysis of strongly nonlinear PDEs, Acta Numer., 26 (2017), pp. 137-303, https://doi.org/10.1017/S0962492917000071 · Zbl 1381.65092 |
| [22] | R. H. Nochetto, D. Ntogakas, and W. Zhang, Two-scale method for the Monge-Ampére equation: pointwise error estimates, IMA J. Numer. Anal., 39 (2019), pp. 1085-1109. · Zbl 1466.65198 |
| [23] | A. Oberman, Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), pp. 221-238. · Zbl 1145.65085 |
| [24] | M. V. Safonov, Nonuniqueness for second-order elliptic equations with measurable coefficients, SIAM J. Math. Anal., 30 (1999), pp. 879-895. · Zbl 0924.35004 |
| [25] | A. J. Salgado and W. Zhang, Finite element approximation of the Isaacs equation, ESAIM Math. Model. Numer. Anal., 53 (2019), pp. 351-374. · Zbl 1433.65311 |
| [26] | I. Smears and E. Süli, Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cordes coefficients, SIAM J. Numer. Anal. 52 (2014), pp. 993-1016. · Zbl 1304.65252 |
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