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A Hybrid Finite Difference WENO-ZQ Fast Sweeping Method for Static Hamilton–Jacobi Equations

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Abstract

In this paper, we propose to combine a new fifth order finite difference weighted essentially non-oscillatory (WENO) scheme with high order fast sweeping methods, for directly solving static Hamilton–Jacobi equations. This is motivated by the work in Xiong et al. (J Sci Comput 45(1–3):514–536, 2010), where a fifth order fast sweeping method base on the classical finite difference WENO scheme is developed. Numerical results in Xiong et al. (2010) show that the iterative numbers of the scheme for some cases are very sensitive to the parameter \(\epsilon \), which is used to avoid the denominator to be 0 in the nonlinear weights. Here we propose to use the new fifth order finite difference WENO-ZQ scheme, which was recently developed in Zhu and Qiu (J Comput Phys 318:110–121, 2016), to alleviate this problem. Besides, to save computational cost from WENO reconstructions, a hybrid finite difference linear and WENO scheme is used, which works more robustly. Numerical experiments will be performed to demonstrate the good performance of the new proposed approach.

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References

  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(3), 667–695 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. Fomel, S., Luo, S., Zhao, H.K.: Fast sweeping method for the factored eikonal equation. J. Comput. Phys. 228, 6440–6455 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Helmsen, J., Puckett, E., Colella, P., Dorr, M.: Two new methods for simulating photolithography development in 3D. Proc. SPIE 2726, 253–262 (1996)

    Article  Google Scholar 

  5. Huang, L., Shu, C.-W., Zhang, M.P.: Numerical boundary conditions for the fast sweeping high order WENO methods for solving the Eikonal equation. J. Comput. Math. 26(3), 336–346 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Huang, L., Wong, S.C., Zhang, M., Shu, C.-W., Lam, W.H.K.: Revisiting Hughes’ dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm. Transp. Res. B-Meth. 43(1), 127–141 (2009)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kao, C.-Y., Osher, S., Qian, J.: Lax–Friedrichs sweeping scheme for static Hamilton–Jacobi equations. J. Comput. Phys. 196(1), 367–391 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kao, C.-Y., Osher, S., Tsai, Y.H.: Fast sweeping methods for static Hamilton–Jacobi equations. SIAM J. Numer. Anal. 42(6), 2612–2632 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Levy, D., Puppo, G., Russo, G.: Central WENO schemes for hyperbolic systems of conservation laws, M2AN. Math. Model. Numer. Anal. 33(3), 547–571 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Levy, D., Puppo, G., Russo, G.: Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22(2), 656–672 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, F., Shu, C.-W., Zhang, Y.-T., Zhao, H.: Second order discontinuous Galerkin fast sweeping method for Eikonal equations. J. Comput. Phys. 227(17), 8191–8208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luo, S.: A uniformly second order fast sweeping method for Eikonal equations. J. Comput. Phys. 241(10), 104–117 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin, J., Abgrall, R., Qiu, J.: High order residual distribution for steady state problems for hyperbolic conservation laws. J. Sci. Comput 79(2), 891–913 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Qian, J., Cheng, L.T., Osher, S.: A level set based Eulerian approach for anisotropic wave propagations. Wave. Motion. 37(4), 365–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Qian, J., Zhang, Y.-T., Zhao, H.-K.: A fast sweeping method for static convex Hamilton–Jacobi equations. J. Sci. Comput. 31(1), 237–271 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Serna, S., Qian, J.: A stopping criterion for higher-order sweeping schemes for static Hamilton–Jacobi equations. J. Comput. Math. 28(4), 552–568 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Nat. Acad. Sci. 93(4), 1591–1595 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shu, C.-W.: High order numerical methods for time dependent Hamilton–Jacobi equations. Math. Comput. Imaging Sci. Inf. Process. (2007)

  22. Tan, S., Shu, C.-W.: Inverse Lax–Wendroff procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229(21), 8144–8166 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tsai, R., Cheng, L.T., Osher, S., Zhao, H.-K.: Fast sweeping algorithms for a class of Hamilton–Jacobi equations. SIAM J. Numer. Anal. 41(2), 673–694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Autom. Contr. 40(9), 1528–1538 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Versteeg, R.: The Marmousi experience: velocity model determination on a synthetic complex data set. Lead. Edge 13(09), 927–936 (1994)

    Article  Google Scholar 

  26. Wu, L., Zhang, Y.-T.: A third order fast sweeping method with linear computational complexity for Eikonal equations. J. Sci. Comput. 62(1), 198–229 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xia, Y., Wong, S.C., Zhang, M., Shu, C.-W., Lam, W.H.K.: An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model. Int. J. Numer. Meth. Eng. 76(3), 337–350 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Xiong, T., Zhang, M.P., Zhang, Y.-T., Shu, C.-W.: Fast sweeping fifth order WENO scheme for static Hamilton–Jacobi equations with accurate boundary treatment. J. Sci. Comput. 45(1–3), 514–536 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, Y.-T., Chen, S., Li, F., Zhao, H.-K., Shu, C.-W.: Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations. SIAM J. Sci. Comput. 33(4), 1873–1896 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, Y.-T., Zhao, H.-K., Qian, J.: High Order fast sweeping methods for static Hamilton–Jacobi equations. J. Sci. Comput. 29(1), 25–56 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhao, H.-K.: A fast sweeping method for Eikonal equations. Math. Comput. 74(250), 603–627 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhao, H.-K., Osher, S., Merriman, B., Kang, M.: Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method. Comput. Vis. Image. Und. 80(3), 295–314 (2000)

    Article  MATH  Google Scholar 

  33. Zhao, Z., Zhu, J., Chen, Y., Qiu, J.: A new hybrid WENO scheme for hyperbolic conservation laws. Comput. Fluids. 179, 422–436 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for Hamilton–Jacobi equations. Numer. Meth. Part. D. E. 33(4), 1095–1113 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhu, J., Qiu, J.: A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110–121 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhu, J., Qiu, J.: A new type of finite volume WENO schemes for hyperbolic conservation laws 73(2–3), 1338–1359 (2017)

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Yupeng Ren

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Tao Xiong & Jianxian Qiu

Authors
  1. Yupeng Ren
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  2. Tao Xiong
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  3. Jianxian Qiu
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Correspondence to Jianxian Qiu.

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The research is partly supported by NSAF Grant U1630247, Science Challenge Project, No. TZ2016002, NSFC Grant 11971025, NSF Grant of Fujian Province 2019J06002 and Sino-German Research Group Project, No. GZ. 1465.

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Ren, Y., Xiong, T. & Qiu, J. A Hybrid Finite Difference WENO-ZQ Fast Sweeping Method for Static Hamilton–Jacobi Equations. J Sci Comput 83, 54 (2020). https://doi.org/10.1007/s10915-020-01228-7

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  • DOI: https://doi.org/10.1007/s10915-020-01228-7

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