Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T13:06:26.968Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Volume Hermite WENO Schemes for Solving the Hamilton-Jacobi Equation

Published online by Cambridge University Press:  03 June 2015

Jun Zhu  and
Jianxian Qiu
Jun Zhu*
Affiliation:
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Jiangsu, China
Jianxian Qiu*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, China
*
Email:zhujun@nuaa.edu.cn
Corresponding author.Email:jxqiu@xmu.edu.cn
Get access

Abstract

In this paper, we present a new type of Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the Hamilton-Jacobi equations on the finite volume framework. The cell averages of the function and its first one (in one dimension) or two (in two dimensions) derivative values are together evolved via time approaching and used in the reconstructions. And the major advantages of the new HWENO schemes are their compactness in the spacial field, purely on the finite volume framework and only one set of small stencils is used for different type of the polynomial reconstructions. Extensive numerical tests are performed to illustrate the capability of the methodologies.

Keywords

65M0665M9935L65HWENO schemefinite volumeHamilton-Jacobi equation
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 959 - 980
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abgrall, R. and Sonar, T., On the use of Muehlbach expansions in the recovery step of ENO methods, Numer. Math., 76 (1997), 1–25.CrossRefGoogle Scholar
[2]Augoula, S. and Abgrall, R., High order numerical discretization for Hamilton-Jacobi equations on triangular meshes, J. Sci. Comput., 15 (2000), 198–229.CrossRefGoogle Scholar
[3]Barth, T. J. and Sethian, J. A., Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comput. Phys., 145 (1998), 1–40.Google Scholar
[4]Chan, C. K., Lau, K. S. and Zhang, B. L., Simulation of a premixed turbulent Name with the discrete vortex method, Int. J. Numer. Meth. Eng., 48 (2000), 613–627.3.0.CO;2-7>CrossRefGoogle Scholar
[5]Cheng, Y. D. and Shu, C.-W., A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, J. Comput. Phys., 223 (2007), 398–415.Google Scholar
[6]Hu, C. and Shu, C.-W., A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (1999), 666–690.CrossRefGoogle Scholar
[7]Jiang, G. S. and Peng, D., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (2000), 2126–2143.CrossRefGoogle Scholar
[8]Jiang, G. S. and Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228.Google Scholar
[9]Jin, S. and Xin, Z., Numerical passage from systems of conservation laws to Hamilton-Jacobi equations, and relaxation schemes, SIAM J. Numer. Anal., 35 (1998), 2163–2186.CrossRefGoogle Scholar
[10]Lafon, F. and Osher, S., High order two dimensional nonoscillatory methods for solving Hamilton-Jacobi scalar equations, J. Comput. Phys., 123 (1996), 235–253.Google Scholar
[11]Lepsky, O., Hu, C. and Shu, C.-W., Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations, Appl. Numer. Math., 33 (2000), 423–434.Google Scholar
[12]Li, X. G. and Chan, C. K., High-order schemes for Hamilton-Jacobi equations on triangular meshes, J. Comput. Appl. Math., 167 (2004), 227–241.Google Scholar
[13]Lions, P. L., Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London, 1982.Google Scholar
[14]Osher, S. and Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12–49.Google Scholar
[15]Osher, S. and Shu, C.-W., High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), 907–922.Google Scholar
[16]Qiu, J., WENO schemes with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations, J. Comput. Appl. Math., 200 (2007), 591–605.Google Scholar
[17]Qiu, J., Hermite WENO schemes with Lax-Wendroff type time discretizations for Hamilton-Jacobi equations, J. Comput. Math., 25 (2007), 131–144.Google Scholar
[18]Qiu, J. and Shu, C.-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one dimensional case, J. Comput. Phys., 193 (2004), 115–135.CrossRefGoogle Scholar
[19]Qiu, J. and Shu, C.-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: two-dimensional case, Comput. Fluids, 34 (2005), 642–663.Google Scholar
[20]Qiu, J. and Shu, C.-W., Hermite WENO schemes for Hamilton-Jacobi equations, J. Comput. Phys., 204 (2005), 82–99.CrossRefGoogle Scholar
[21]Sethian, J. A., Level Set Methods and Fast Marching Methods, Evolving Interface, Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge, 1999.Google Scholar
[22]Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. edited by Quarteroni, A., Editor, Lecture Notes in Mathematics, CIME subseries (Springer-Verlag, Berlin/New York); ICASE Report 97–65, 1997.Google Scholar
[23]Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77 (1988), 439–471.CrossRefGoogle Scholar
[24]Yan, J. and Osher, S., A local discontinuous Galerkin method for directly solving Hamilton-Jacobi equations, J. Comput. Phys., 230 (2011), 232–244.Google Scholar
[25]Zhang, Y. T. and Shu, C.-W., High-order WENO schemes for Hamilton-Jacobi equations on triangular meshes, SIAM J. Sci. Comput., 24 (2003), 1005–1030.CrossRefGoogle Scholar
[26]Zhu, J. and Qiu, J., A class of the fourth order finite volume Hermite weighted essentially non-oscillatory schemes, Science in China, Series A-Mathematics, 51 (2008), 1549–1560.Google Scholar
[27]Zhu, J. and Qiu, J., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method III: unstructured meshes, J. Sci. Comput., 39 (2009), 293–321.Google Scholar