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Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs

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Abstract

This paper approximates viscosity solutions of fully nonlinear second order parabolic PDEs by a narrow-stencil finite difference spatial-discretization paired with the forward Euler method for the time discretization. Using generalized monotonicity of the numerical scheme and an iteration technique, we prove the numerical scheme is stable in both the \(l^2\)-norm and the \(l^{\infty }\)-norm. Then under this stability, we establish the convergence of the proposed numerical scheme to the viscosity solution of the underlying fully nonlinear second order parabolic PDEs based on different regularities with respect to time and space. Finally, we report some numerical experiments.

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References

  1. Barles, G., Perthame, B.: Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control. Optim. 26(5), 1133–1148 (1988)

    Article  MathSciNet  Google Scholar 

  2. Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4(3), 271–283 (1991)

    MathSciNet  Google Scholar 

  3. Caffarelli, L.A., Cabré, X., Edmunds, D.E.: Fully nonlinear elliptic equations, Providence, RI: American Mathematical. Society. 43, vi–104 (1995)

  4. Camilli, F., Falcone, M.: An approximation scheme for the optimal control of diffusion processes. ESAIM: Math. Modell. Numer. Anal. 29(1), 97–122 (1995)

    Article  MathSciNet  Google Scholar 

  5. Camilli, F., Jakobsen, E.R.: A finite element like scheme for integro-partial differential Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal. 47(4), 2407–2431 (2009)

    Article  MathSciNet  Google Scholar 

  6. Ciarlet, P.G.: Linear and nonlinear functional analysis with applications. Siam. 130 (2013)

  7. Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)

    Article  MathSciNet  Google Scholar 

  8. Debrabant, K., Jakobsen, E.R.: Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comput. 82(283), 1433–1426 (2013)

    Article  MathSciNet  Google Scholar 

  9. Feng, X., Glowinski, R., Neilan, M.: Recent developments in numerical methods for fully nonlinear second order partial differential equations. Siam Rev. 55(2), 205–267 (2013)

    Article  MathSciNet  Google Scholar 

  10. Feng, X., Jensen, M.: Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids. SIAM J. Numer. Anal. 55(2), 691–712 (2017)

    Article  MathSciNet  Google Scholar 

  11. Feng, X., Kao, C.Y., Lewis, T.: Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations. J. Comput. Appl. Math. 254, 81–98 (2013)

    Article  MathSciNet  Google Scholar 

  12. Feng, X., Lewis, T.: Nonstandard local discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions. J. Sci. Comput. 77(3), 1534–1565 (2018)

    Article  MathSciNet  Google Scholar 

  13. Feng, X., Lewis, T.: A Narrow-stencil finite difference method for approximating viscosity solutions of fully nonlinear elliptic partial differential equations with applications to Hamilton-Jacobi-Bellman equations. arXiv preprint arXiv:1907.10204 (2019)

  14. Feng, X., Lewis, T.: A narrow-stencil finite difference method for approximating viscosity solutions of Hamilton-Jacobi-Bellman Equations. SIAM J. Numer. Anal. 59(2), 886–924 (2021)

    Article  MathSciNet  Google Scholar 

  15. Feng, X., Lewis, T., Neilan, M.: Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations. J. Comput. Appl. Math. 299, 68–91 (2016)

    Article  MathSciNet  Google Scholar 

  16. Feng, X., Lewis, T., Rapp, A.: Dual-wind discontinuous Galerkin methods for stationary Hamilton-Jacobi equations and regularized Hamilton-Jacobi equations. Commun. Appl. Math. Comput. 4(2), 563–596 (2022)

    Article  MathSciNet  Google Scholar 

  17. Feng, X., Lewis, T., Ward, K.: A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs. In: 2021 UNC Greensboro PDE Conference, Electronic Journal of Difference Equations. Conference, vol. 26, pp. 59–95 (2022)

  18. Feng, X., Neilan, M.: Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47(2), 1226–1250 (2009)

    Article  MathSciNet  Google Scholar 

  19. Feng, X., Neilan, M.: Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J. Sci. Comput. 38(1), 74–98 (2009)

    Article  MathSciNet  Google Scholar 

  20. Feng, X., Neilan, M.: Finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method. Comput. Math. Appl. 68(12), 2182–2204 (2014)

    Article  MathSciNet  Google Scholar 

  21. Fleming, W.H., Rishel, R.W.: Deterministic and stochastic optimal control. vol. 1, Springer Science & Business Media, Berlin (2012)

  22. Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solutions. vol. 25, Springer Science & Business Media, Berlin (2006)

  23. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. vol. 224, No. (2), Springer, Berlin (1977)

  24. Jensen, M., Smears, I.: On the convergence of finite element methods for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal. 51(1), 137–162 (2013)

    Article  MathSciNet  Google Scholar 

  25. Kushner, H.J.: Numerical methods for stochastic control problems in continuous time. SIAM J. Control. Optim. 28(5), 999–1048 (1990)

    Article  MathSciNet  Google Scholar 

  26. Lewis, T.L.: Finite difference and discontinuous Galerkin finite element methods for fully nonlinear second order partial differential equations. Ph.D. Thesis, University of Tennessee (2013)

  27. Liu, H., Yamamoto, M.: Stability in determination of states for the mean field game equations. Commun. Anal. Comput. 1(2), 157–167 (2023)

    Google Scholar 

  28. Nochetto, R.H., Ntogkas, D., Zhang, W.: Two-scale method for the Monge-Ampère equation: convergence to the viscosity solution. Math. Comput. 88(316), 637–664 (2019)

    Article  Google Scholar 

  29. Nochetto, R.H., Ntogakas, D., Zhang, W.: Two-scale method for the Monge-Ampère equation: pointwise error estimates. IMA J. Numer. Anal. 39(3), 1085–1109 (2019)

    Article  MathSciNet  Google Scholar 

  30. Oberman, A.M.: Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discret. Contin. Dyn. Syst. B 10(1), 221 (2008)

    Google Scholar 

  31. Shkoller, S.: Notes on Lp and Sobolev spaces. Web site: http://www.math.ucdavis.edu/hunter/m218b, 9 (2009)

  32. Wang, L., Wei, T., Zheng, G.: Determining the random source and initial value simultaneously in stochastic fractional diffusion equations. Commun. Anal. Comput. 1(3), 234–270 (2023)

    Google Scholar 

  33. Zhang, C., Wang, Y., Zhang, X.: Operator-valued martingale transform on mixed martingale Hardy spaces. Commun. Anal. Comput. 1(4), 369–378 (2023)

    Google Scholar 

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Funding

Weifeng Qiu is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302219, CityU 11300621).

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  1. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

    Xiang Zhong & Weifeng Qiu

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  1. Xiang Zhong
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Xiang Zhong and Weifeng Qiu have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, method, analysis and writing. All authors certify that this material or similar material has not been and will not be submitted to or published in any other publication.

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Zhong, X., Qiu, W. Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs. J Sci Comput 99, 76 (2024). https://doi.org/10.1007/s10915-024-02544-y

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  • DOI: https://doi.org/10.1007/s10915-024-02544-y

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