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High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations

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Abstract

We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton–Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss–Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  • Augoula S., Abgrall R. (2000). High order numerical discretization for Hamilton–Jacobi equations on triangular meshes. J. Sci. Comput. 15, 197–229

    Article  MATH  MathSciNet  Google Scholar 

  • Boué M., Dupuis P. (1999). Markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control. SIAM J. Numer. Anal. 36, 667–695

    Article  MathSciNet  Google Scholar 

  • Bryson S.,Levy D. (2003). High-order central WENO schemes for multidimensional Hamilton–Jacobi equations. SIAM J Numer. Anal. 41, 1339–1369

    Article  MATH  MathSciNet  Google Scholar 

  • Barth T., Sethian J. (1998). Numerical schemes for the Hamilton–Jacobi and level set equations on triangulated domains J. Comput. Phys. 145, 1–40

    Article  MATH  MathSciNet  Google Scholar 

  • Cecil T., Qian J., Osher S. (2004). Numerical methods for high dimensional Hamilton–Jacobi equations using radial basis functions. J. Comput. Phys. 196, 327–347

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Dellinger, J., and Symes, W. W. (1997). Anisotropic finite-difference traveltimes using a Hamilton–Jacobi solver, 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1786–1789

  • Falcone M., Ferretti R. (1994). Discrete time high-order schemes for viscosity solutions of Hamilton–Jacobi–Bellman equations Numer. Math. 67, 315–344

    MATH  MathSciNet  Google Scholar 

  • Falcone M., Ferretti R. (2002). Semi-Lagrangian schemes for Hamilton–Jacobi equations, discrete representation formulae and Godunov methods. J. Comput. Phys. 175, 559–575

    Article  MATH  MathSciNet  Google Scholar 

  • Gray S., May W. (1994). Kirchhoff migration using eikonal equation travel-times. Geophysics 59, 810-817

    Article  Google Scholar 

  • Helmsen J., Puckett E., Colella P., Dorr M. (1996) Two new methods for simulating photolithography development in 3d, Proc SPIE, 2726: 253–261

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kim S., Cook R. (1999). 3D traveltime computation using second-order ENO scheme. Geophys. 64, 1867–1876

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Lin C.-T., Tadmor E. (2000). High-resolution non-oscillatory central schemes for approximate Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21, 2163–2186

    Article  MATH  MathSciNet  Google Scholar 

  • Lin C.-T., Tadmor E. (2001). L 1-stability and error estimates for approximate Hamilton–Jacobi solutions. Numer Math. 87, 701–735

    Article  MATH  MathSciNet  Google Scholar 

  • Osher S. (1993). A level set formulation for the solution of the Dirichlet problem for Hamilton–Jacobi equations. SIAM J. Math Anal. 24, 1145–1152

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Peng D., Osher S., Merriman B., Zhao H.-K., Rang M. (1999). A PDE-based fast local level set method. J. Comput Phys. 155, 410–438

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Qian J., Symes W.W. (2001). Paraxial eikonal solvers for anisotropic quasi-P travel times. J. Comput Phys. 174, 256–278

    Article  Google Scholar 

  • Qian J., Symes W.W. (2002). Finite-difference quasi-P traveltimes for anisotropic media. Geophysics 67, 147–155

    Article  Google Scholar 

  • Qian J., Symes W.W. (2002). An adaptive finite-difference method for traveltime and amplitude. Geophysics 67, 166–176

    Google Scholar 

  • Qin F., Schuster G. T. (1993). First-arrival traveltime calculation for anisotropic media. Geophysics 58, 1349–1358

    Article  Google Scholar 

  • Rouy E., Tourin A. (1992). A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29, 867–884

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  PubMed  MATH  MathSciNet  Google Scholar 

  • Sethian J.A., Vladimirsky A. (2001). Ordered upwind methods for static Hamilton–Jacobi equations. Proc. Natl. Acad. Sci. 98, 11069–11074

    Article  PubMed  MATH  MathSciNet  Google Scholar 

  • Sethian J.A., Vladimirsky A. (2003) Ordered upwind methods for static Hamilton–Jacobi equations: theory and algorithms SIAM J. Numer. Anal. 41, 325–363

    Article  MATH  MathSciNet  Google Scholar 

  • Chi-Wang Shu (2004) High Order Numerical Methods for Time Dependent Hamilton–Jacobi Equations, WSPC/Lecture Notes Series

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Tsitsiklis J.N. (1995). Efficient algorithms for globally optimal trajectories. IEEE Trans. Auto. Control 40, 1528–1538

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Zhao H. (2004). A fast sweeping method for Eikonal equations. Math Comp. 74, 603–627

    Article  Google Scholar 

  • Zhao H., Osher S., Merriman B., Kang M. (2000). Implicit and non-parametric shape reconstruction from unorganized points using variational level set method. Comput. Vision Image Understand. 80, 295–319

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, CA, 92697-3875, USA

    Yong-Tao Zhang & Hong-Kai Zhao

  2. Department of Mathematics, University of California, Los Angeles, CA, 90095-1555, USA

    Jianliang Qian

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  1. Yong-Tao Zhang
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  2. Hong-Kai Zhao
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  3. Jianliang Qian
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Correspondence to Yong-Tao Zhang or Hong-Kai Zhao.

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Zhang, YT., Zhao, HK. & Qian, J. High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations. J Sci Comput 29, 25–56 (2006). https://doi.org/10.1007/s10915-005-9014-3

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  • DOI: https://doi.org/10.1007/s10915-005-9014-3

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