Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-07T12:11:02.477Z Has data issue: false hasContentIssue false
  • English
  • Français

An h-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

Published online by Cambridge University Press:  28 May 2015

Hongqiang Zhu  and
Jianxian Qiu
Hongqiang Zhu*
Affiliation:
School of Natural Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu, China
Jianxian Qiu*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

In [35,36], we presented an h-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws. A tree data structure (binary tree in one dimension and quadtree in two dimensions) is used to aid storage and neighbor finding. Mesh adaptation is achieved by refining the troubled cells and coarsening the untroubled “children”. Extensive numerical tests indicate that the proposed h-adaptive method is capable of saving the computational cost and enhancing the resolution near the discontinuities. In this paper, we apply this h-adaptive method to solve Hamilton-Jacobi equations, with an objective of enhancing the resolution near the discontinuities of the solution derivatives. One- and two-dimensional numerical examples are shown to illustrate the capability of the method.

Keywords

65M6065M9935L65Runge-Kutta discontinuous Galerkin methodh-adaptive methodHamilton-Jacobi equation
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 4 , November 2013 , pp. 617 - 636
Copyright
Copyright © Global Science Press Limited 2013

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]Albert, S., Cockburn, B., French, D., and Peterson, T., A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations, part I: the steady state case, Math. Comput., 71 (2001), pp. 49–76.CrossRefGoogle Scholar
[2]Bryson, S. and Levy, D., High-order central WENO schemes for multidimensional Hamilton-Jacobi equations, SIAM J. Numer. Anal., 41 (2003), pp. 1339–1369.CrossRefGoogle Scholar
[3]Bryson, S. and Levy, D., High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton-Jacobi equations, J. Comput. Phys., 189 (2003), pp. 63–87.CrossRefGoogle Scholar
[4]Chen, Y. and Cockburn, B., An adaptive high-order discontinuous Galerkin method with error control for the Hamilton-Jacobi equations, part I: the one-dimensional steady state case, J. Comput. Phys., 226 (2007), pp. 1027–1058.CrossRefGoogle Scholar
[5]Cheng, Y. and Shu, C.-W., A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations, J. Comput. Phys., 223 (2007), pp. 398–415.CrossRefGoogle Scholar
[6]Cockburn, B., Hou, S., and Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54 (1990), pp. 545–581.Google Scholar
[7]Cockburn, B., Lin, S.-Y., and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J. Comput. Phys., 84 (1989), pp. 90–113.CrossRefGoogle Scholar
[8]Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52 (1989), pp. 411–435.Google Scholar
[9]Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199–224.CrossRefGoogle Scholar
[10]Cockburn, B. and Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), pp. 173–261.CrossRefGoogle Scholar
[11]Crandall, M. G. and Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), pp. 1–42.CrossRefGoogle Scholar
[12]Crandall, M. G. and Lions, P. L., Two approximations of solutions of Hamilton-Jacobi equations, Math. Comput., 43 (1984), pp. 1–19.CrossRefGoogle Scholar
[13]Harten, A., Engquist, B., Osher, S., and Chakravathy, S., Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), pp. 231–303.CrossRefGoogle Scholar
[14]Hu, C. and Shu, C.-W., A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (1999), pp. 666–690.CrossRefGoogle Scholar
[15]Jiang, G. and Peng, D., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (2000), pp. 2126–2143.CrossRefGoogle Scholar
[16]Jiang, G. and Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), pp. 202–228.CrossRefGoogle Scholar
[17]Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., and Flaherty, J., Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math., 48 (2004), pp. 323–338.CrossRefGoogle Scholar
[18]Kurganov, A. and Tadmor, E., New high-resolution semi-discrete central schemes for Hamilton- Jacobi equations, J. Comput. Phys., 160 (2000), pp. 720–742.CrossRefGoogle Scholar
[19]Li, F. and Shu, C.-W., Reinterpretation and simplified implementation of a discontinuous Galerkin method for Hamilton-Jacobi equations, Appl. Math. Lett., 18 (2005), pp. 1204–1209.CrossRefGoogle Scholar
[20]Lin, C.T. and Tadmor, E., High-resolution non-oscillatory central schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21 (2000), pp. 2163–2186.CrossRefGoogle Scholar
[21]Liu, X.-D., Osher, S., and Chan, T., Weighted essentially nonoscillatory schemes, J. Comput. Phys., 115 (1994), pp. 200–212.CrossRefGoogle Scholar
[22]Osher, S. and Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12–49.CrossRefGoogle Scholar
[23]Osher, S. and Shu, C.-W., High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28 (1991), pp. 907–922.CrossRefGoogle Scholar
[24]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), pp. 115–135.CrossRefGoogle Scholar
[25]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. Fluid., 34 (2005), pp. 642–663.CrossRefGoogle Scholar
[26]Qiu, J. and Shu, C.-W., Hermite WENO schemes for Hamilton-Jacobi equations, J. Comput. Phys., 204 (2005), pp. 82–99.CrossRefGoogle Scholar
[27]Qiu, J. and Shu, C.-W., Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput., 26 (2005), pp. 907–929.CrossRefGoogle Scholar
[28]Reed, W. H. and Hill, T. R., Triangular mesh methods for neutron transport equation, Technical report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.Google Scholar
[29]Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Math. Comput., 49 (1987), pp. 105–121.CrossRefGoogle Scholar
[30]Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439–471.CrossRefGoogle Scholar
[31]Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, >II, J. Comput. Phys., 83 (1989), pp. 32–78.CrossRefGoogle Scholar
[32]Tang, H. Z., Tang, T., and ZHANG, P W, An adaptive mesh redistribution method for nonlinear Hamilton-Jacobi equations in two- and three-dimensions, J. Comput. Phys., 188 (2003), pp. 543–572.CrossRefGoogle Scholar
[33]Yan, J. and Osher, S., A local discontinuous Galerkin method for directly solving Hamilton-Jacobi equations, J. Comput. Phys., 230 (2011), pp. 232–244.CrossRefGoogle Scholar
[34]Zhong, X. and Shu, C.-W., A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods, J. Comput. Phys., 232 (2013), pp. 397–415.CrossRefGoogle Scholar
[35]Zhu, H. and Qiu, J., Adaptive Runge-Kutta discontinuous Galerkin methods using different indicators: one-dimensional case, J. Comput. Phys., 228 (2009), pp. 6957–6976.CrossRefGoogle Scholar
[36]Zhu, H. and Qiu, J., An h-adaptive RKDG method with troubled-cell indicator for two-dimensional hyperbolic conservation laws, Adv. Comput. Math., DOI: 10.1007/s10444-012-9287-7.Google Scholar