Journal of Scientific Computing

, Volume 62, Issue 1, pp 198–229 | Cite as

A Third Order Fast Sweeping Method with Linear Computational Complexity for Eikonal Equations

Article

Abstract

Fast sweeping methods are a class of efficient iterative methods for solving steady state hyperbolic PDEs. They utilize the Gauss-Seidel iterations and alternating sweeping strategy to cover a family of characteristics of the hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The first order fast sweeping method for solving Eikonal equations (Zhao in Math Comput 74:603–627, 2005) has linear computational complexity, namely, the computational cost is \(O(N)\) where \(N\) is the number of grid points of the computational mesh. Recently, a second order fast sweeping method with linear computational complexity was developed in Zhang et al. (SIAM J Sci Comput 33:1873–1896, 2011). The method is based on a discontinuous Galerkin (DG) finite element solver and causality indicators which guide the information flow directions of the nonlinear Eikonal equations. How to extend the method to higher order accuracy is still an open problem, due to the difficulties of solving much more complicated local nonlinear systems and calculations of local causality information. In this paper, we extend previous work and develop a third order fast sweeping method with linear computational complexity for solving Eikonal equations. A novel approach is designed for capturing the causality information in the third order DG local solver. Numerical experiments show that the method has third order accuracy and a linear computational complexity.

Keywords

Fast sweeping methods Discontinuous Galerkin methods  High order accuracy Linear computational complexity Static Hamilton–Jacobi equations  Eikonal equations 

Mathematics Subject Classification

65M99 

References

  1. 1.
    Boué, M., Dupuis, P.: Markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control. SIAM J. Numer. Anal. 36, 667–695 (1999)CrossRefMathSciNetGoogle Scholar
  2. 2.
    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)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Crandall, M.G., Lions, P.L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277, 1–42 (1983)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Dijkstra, E.W.: A note on two problems in connection with graphs. Numer. Math. 1, 269–271 (1959)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Fomel, S., Luo, S., Zhao, H.: Fast sweeping method for the factored eikonal equation. J. Comput. Phys. 228, 6440–6455 (2009)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Helmsen, J., Puckett, E., Colella, P., Dorr, M.: Two new methods for simulating photolithography development in 3D. Proc. SPIE 2726, 253–261 (1996)CrossRefGoogle Scholar
  7. 7.
    Hu, C., Shu, C.-W.: A discontinuous Galerkin finite element method for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 20, 666–690 (1999)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Huang, L., Shu, C.-W., Zhang, M.: Numerical boundary conditions for the fast sweeping high order WENO methods for solving the Eikonal equation. J. Comput. Math. 26, 336–346 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Kao, C.Y., Osher, S., Qian, J.: Lax–Friedrichs sweeping schemes for static Hamilton–Jacobi equations. J. Comput. Phys. 196, 367–391 (2004)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Kao, C.Y., Osher, S., Qian, J.: Legendre-transform-based fast sweeping methods for static Hamilton–Jacobi equations on triangulated meshes. J. Comput. Phys. 227, 10209–10225 (2008)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Li, F., Shu, C.-W., Zhang, Y.-T., Zhao, H.-K.: A second order discontinuous Galerkin fast sweeping method for Eikonal equations. J. Comput. Phys. 227, 8191–8208 (2008)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Qian, J., Zhang, Y.-T., Zhao, H.-K.: Fast sweeping methods for Eikonal equations on triangular meshes. SIAM J. Numer. Anal. 45, 83–107 (2007)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Qian, J., Zhang, Y.-T., Zhao, H.-K.: A fast sweeping method for static convex Hamilton–Jacobi equations. J. Sci. Comput. 31, 237–271 (2007)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Rouy, E., Tourin, A.: A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29, 867–884 (1992)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Serna, S., Qian, J.: A stopping criterion for higher-order sweeping schemes for static Hamilton–Jacobi equations. J. Comput. Math. 28, 552–568 (2010)MathSciNetMATHGoogle Scholar
  17. 17.
    Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. USA 93, 1591–1595 (1996)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Sethian, J.A., Vladimirsky, A.: Ordered upwind methods for static Hamilton–Jacobi equations. Proc. Natl. Acad. Sci. USA 98, 11069–11074 (2001)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Sethian, J.A., Vladimirsky, A.: Ordered upwind methods for static Hamilton–Jacobi equations: theory and algorithms. SIAM J. Numer. Anal. 41, 325–363 (2003)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Tan, S., Shu, C.-W.: Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229, 8144–8166 (2010)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Tsai, Y.-H., Cheng, L.-T., Osher, S., Zhao, H.-K.: Fast sweeping algorithms for a class of Hamilton–Jacobi equations. SIAM J. Numer. Anal. 41, 673–694 (2003)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Autom. Control 40, 1528–1538 (1995)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Xiong, T., Zhang, M., Zhang, Y.-T., Shu, C.-W.: Fifth order fast sweeping WENO scheme for static Hamilton–Jacobi equations with accurate boundary treatment. J. Sci. Comput. 45, 514–536 (2010)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Zhang, Y.-T., Chen, S., Li, F., Zhao, H., Shu, C.-W.: Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations. SIAM J. Sci. Comput. 33, 1873–1896 (2011)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    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)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Zhang, Y.-T., Zhao, H.-K., Chen, S.: Fixed-point iterative sweeping methods for static Hamilton–Jacobi equations. Methods Appl. Anal. 13, 299–320 (2006)MathSciNetMATHGoogle Scholar
  27. 27.
    Zhang, Y.-T., Zhao, H.-K., Qian, J.: High order fast sweeping methods for static Hamilton–Jacobi equations. J. Sci. Comput. 29, 25–56 (2006)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Zhao, H.-K.: A fast sweeping method for Eikonal equations. Math. Comput. 74, 603–627 (2005)CrossRefMATHGoogle Scholar
  29. 29.
    Zhao, H., Osher, S., Merriman, B., Kang, M.: Implicit and non-parametric shape reconstruction from unorganized points using variational level set method. Comput. Vis. Image Underst. 80, 295–319 (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameUSA

Personalised recommendations