Skip to main content
Springer Nature Link
Log in

An Alternative Formulation of Discontinous Galerkin Schemes for Solving Hamilton–Jacobi Equations

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

Abstract

The aim of this paper is to develop an alternative formulation of discontinuous Galerkin (DG) schemes for approximating the viscosity solutions to nonlinear Hamilton–Jacobi (HJ) equations. The main difficulty in designing DG schemes lies in the inherent non-divergence form of HJ equations. One effective approach is to explore the elegant relationship between HJ equations and hyperbolic conservation laws: the standard DG scheme is applied to solve a conservation law system satisfied by the derivatives of the solution of the HJ equation. In this paper, we consider an alternative approach to directly solving the HJ equations, motivated by a class of successful direct DG schemes by Cheng and Shu (J Comput Phys 223(1):398–415, 2007), Cheng and Wang (J Comput Phys 268:134–153, 2014). The proposed scheme is derived based on the idea from the central-upwind scheme by Kurganov et al. (SIAM J Sci Comput 23(3):707–740, 2001). In particular, we make use of precise information of the local speeds of propagation at discontinuous element interface with the goal of adding adequate numerical viscosity and hence naturally capturing the viscosity solutions. A collection of numerical experiments is presented to demonstrate the performance of the method for solving general HJ equations with linear, nonlinear, smooth, non-smooth, convex, or non-convex Hamiltonians.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Abgrall, R.: Numerical discretization of the first-order Hamilton–Jacobi equation on triangular meshes. Commun. Pure Appl. Math. 49(12), 1339–1373 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for directly solving the Hamilton–Jacobi equations. J. Comput. Phys. 223(1), 398–415 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheng, Y., Wang, Z.: A new discontinuous Galerkin finite element method for directly solving the Hamilton–Jacobi equations. J. Comput. Phys. 268, 134–153 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 282(2), 487–502 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Crandall, M.G., Lions, P.L.: Two approximations of solutions of Hamilton–Jacobi equations. Math. Comput. 43(167), 1–19 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Evans, L.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  9. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, W., Li, F., Qiu, J.: Local-structure-preserving discontinuous Galerkin methods with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations. J. Sci. Comput. 47(2), 239–257 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ha, Y., Kim, C., Yang, H., Yoon, J.: A sixth-order weighted essentially non-oscillatory schemes based on exponential polynomials for Hamilton-Jacobi equations. J. Sci. Comput. 75(3), 1675–1700 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harten, A., Lax, P., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hu, C., Shu, C.-W.: A discontinuous Galerkin finite element method for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(2), 666–690 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jin, S., Xin, Z.: Numerical passage from systems of conservation laws to Hamilton–Jacobi equations, and relaxation schemes. SIAM J. Numer. Anal. 35(6), 2385–2404 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kurganov, A., Lin, C.-T.: On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2(1), 141–163 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Kurganov, A., Noelle, S., Petrova, G.: Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 23(3), 707–740 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kurganov, A., Petrova, G.: Adaptive central-upwind schemes for Hamilton–Jacobi equations with nonconvex Hamiltonians. J. Sci. Comput. 27(1), 323–333 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kurganov, A., Petrova, G., Popov, B.: Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws. SIAM J. Sci. Comput. 29(6), 2381–2401 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160(1), 241–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kurganov, A., Tadmor, E.: Solution of two-dimensional riemann problems for gas dynamics without riemann problem solvers. Numer. Methods Partial Differ. Equ. 18(5), 584–608 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lafon, F., Osher, S.: High order two dimensional nonoscillatory methods for solving Hamilton–Jacobi scalar equations. J. Comput. Phys. 123(2), 235–253 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lepsky, O., Hu, C., Shu, C.-W.: Analysis of the discontinuous Galerkin method for Hamilton–Jacobi equations. Appl. Numer. Math. 33(1–4), 423–434 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, F., Shu, C.-W.: Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton–Jacobi equations. Appl. Math. Lett. 18(11), 1204–1209 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, F., Yakovlev, S.: A central discontinuous Galerkin method for Hamilton–Jacobi equations. J. Sci. Comput. 45(1–3), 404–428 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations, vol. 69. Pitman, London (1982)

    MATH  Google Scholar 

  27. Osher, S., Sethian, J.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Osher, S., Shu, C.-W.: High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Qiu, J.: Hermite WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations. J. Comput. Math. 25(2), 131–144 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Qiu, J., Shu, C.-W.: Hermite WENO schemes for Hamilton–Jacobi equations. J. Comput. Phys. 204(1), 82–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shu, C.-W.: High order numerical methods for time dependent Hamilton–Jacobi equations. In: Mathematics and Computation in Imaging Science and Information Processing. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 11, pp. 47–91 (2007)

  32. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tao, Z., Qiu, J.: Dimension-by-dimension moment-based central Hermite WENO schemes for directly solving Hamilton–Jacobi equations. Adv. Comput. Math. 43(5), 1023–1058 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yan, J., Osher, S.: A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations. J. Comput. Phys. 230(1), 232–244 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, Y.-T., Shu, C.-W.: High-order WENO schemes for Hamilton–Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 24(3), 1005–1030 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zheng, F., Qiu, J.: Directly solving the Hamilton–Jacobi equations by Hermite WENO schemes. J. Comput. Phys. 307, 423–445 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zheng, F., Shu, C.-W., Qiu, J.: Finite difference Hermite WENO schemes for the Hamilton–Jacobi equations. J. Comput. Phys. 337, 27–41 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhu, J., Qiu, J.: Hermite WENO schemes for Hamilton–Jacobi equations on unstructured meshes. J. Comput. Phys. 254, 76–92 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 70409, USA

    Guoyi Ke & Wei Guo

Authors
  1. Guoyi Ke
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Wei Guo
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Wei Guo.

Additional information

Research is supported by NSF Grant NSF-DMS-1620047.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ke, G., Guo, W. An Alternative Formulation of Discontinous Galerkin Schemes for Solving Hamilton–Jacobi Equations. J Sci Comput 78, 1023–1044 (2019). https://doi.org/10.1007/s10915-018-0794-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0794-7

Keywords