Skip to main content
SpringerLink
Log in

A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of Hamilton-Jacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a “weighted-residual” or “stabilization-based” formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for Hamilton-Jacobi equations. The L 2 stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general Hamilton-Jacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001/2002)

    Article  MathSciNet  Google Scholar 

  2. Ayuso, B., Marini, L.D.: Discontinuous Galerkin methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 47, 1391–1420 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brezzi, F., Cockburn, B., Marini, L.D., Süli, E.: Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Eng. 195, 3293–3310 (2006)

    Article  MATH  Google Scholar 

  4. Bryson, S., Levy, D.: High-order semi-discrete central-upwind schemes for multidimensional Hamilton-Jacobi equations. J. Comput. Phys. 189, 63–87 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bryson, S., Levy, D.: Mapped WENO and weighted power ENO reconstruction in semi-discrete central schemes for Hamilton-Jacobi equations. Appl. Numer. Math. 56, 1211–1224 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations. J. Comput. Phys. 223, 398–415 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MATH  MathSciNet  Google Scholar 

  8. Cockburn, B., Shu, C.-W.: The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. Math. Model. Numer. Anal. 25, 337–361 (1991)

    MATH  MathSciNet  Google Scholar 

  9. Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cockburn, B., Shu, C.-W.: Special Issue on Discontinuous Galerkin Methods. J. Sci. Comput 22–23 (2005)

  11. Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)

    MATH  MathSciNet  Google Scholar 

  13. Cockburn, B., Karniadakis, G., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G., Shu, C.-W.: Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000)

    Google Scholar 

  14. Cockburn, B., Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194, 588–610 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Crandall, M., Lions, P.L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277, 1–42 (1983)

    MATH  MathSciNet  Google Scholar 

  16. Crandall, M., Lions, P.L.: Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput. 43, 1–19 (1984)

    MATH  MathSciNet  Google Scholar 

  17. Crandall, M.G., Evans, L.C., Lions, P.L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282, 487–502 (1984)

    MATH  MathSciNet  Google Scholar 

  18. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  19. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 214–282 (2000)

    Google Scholar 

  23. Kurganov, A., Tadmor, E.: New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations. J. Comput. Phys. 160, 720–742 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kurganov, A., Noelle, S., Petrova, G.: Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput. 23, 707–740 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lepsky, O., Hu, C., Shu, C.-W.: Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations. Appl. Numer. Math. 33, 423–434 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  27. Li, F., Shu, C.-W.: Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations. Appl. Math. Lett. 18, 1204–1209 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Liu, Y.-J.: Central schemes on overlapping cells. J. Comput. Phys. 209, 82–104 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Liu, Y.-J.: Central schemes and central discontinuous Galerkin methods on overlapping cells. In: Conference on Analysis, Modeling and Computation of PDE and Multiphase Flow, Stony Brook, NY, 2004

  30. Liu, Y.-J., Shu, C.-W., Tadmor, E., Zhang, M.: Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45, 2442–2467 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Liu, Y.-J., Shu, C.-W., Tadmor, E., Zhang, M.: L 2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM, Math. Model. Numer. Anal. 42, 593–607 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  33. Osher, S., Sethian, J.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  34. Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28, 907–922 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  35. Qiu, J., Shu, C.-W.: Hermite WENO schemes for Hamilton-Jacobi equations. J. Comput. Phys. 204, 82–99 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973)

  37. Shu, C.-W.: Total-Variation-Diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988)

    Article  MATH  Google Scholar 

  38. Souganidis, P.E.: Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differ. Equ. 59, 1–43 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  39. Yuan, L., Shu, C.-W.: Discontinuous Galerkin method based on non-polynomial approximation spaces. J. Comput. Phys. 218, 295–323 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zhang, Y.-T., Shu, C.-W.: High-order WENO schemes for Hamilton-Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 24, 1005–1030 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Fengyan Li & Sergey Yakovlev

Authors
  1. Fengyan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey Yakovlev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fengyan Li.

Additional information

The research of F. Li was supported in part by the NSF under the grant DMS-0652481, NSF CAREER award DMS-0847241 and by an Alfred P. Sloan Research Fellowship. Additional support was provided by NSFC grant 10671091 while Li was visiting Department of Mathematics at Nanjing University, China.

The research of S. Yakovlev was supported in part by the NSF CAREER award DMS-0847241.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, F., Yakovlev, S. A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations. J Sci Comput 45, 404–428 (2010). https://doi.org/10.1007/s10915-009-9340-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-009-9340-y

Search

Navigation