Volume 6, Issue 4
An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

Hongqiang Zhu & Jianxian Qiu

Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 617-636.

Published online: 2013-06

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  • 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

Runge-Kutta discontinuous Galerkin method, h-adaptive method, Hamilton-Jacobi equation.

  • AMS Subject Headings

65M60, 65M99, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-6-617, author = {}, title = {An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2013}, volume = {6}, number = {4}, pages = {617--636}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2013.1235nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5922.html} }
TY - JOUR T1 - An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 617 EP - 636 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1235nm UR - https://global-sci.org/intro/article_detail/nmtma/5922.html KW - Runge-Kutta discontinuous Galerkin method, h-adaptive method, Hamilton-Jacobi equation. AB -

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.

Hongqiang Zhu & Jianxian Qiu. (2020). An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations. Numerical Mathematics: Theory, Methods and Applications. 6 (4). 617-636. doi:10.4208/nmtma.2013.1235nm
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