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Volume 29, Issue 5
A High Order Adaptive Finite Element Method for Solving Nonlinear Hyperbolic Conservation Laws

Zhengfu Xu, Jinchao Xu, & Chi-Wang Shu

DOI: 10.4208/jcm.1105-m3392

J. Comp. Math., 29 (2011), pp. 491-500.

Published online: 2011-10

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  • Abstract

In this note, we apply the $h$-adaptive streamline diffusion finite element method with a small mesh-dependent artificial viscosity to solve nonlinear hyperbolic partial differential equations, with the objective of achieving high order accuracy and mesh efficiency. We compute the numerical solution to a steady state Burgers equation and the solution to a converging-diverging nozzle problem. The computational results verify that, by suitably choosing the artificial viscosity coefficient and applying the adaptive strategy based on a posterior error estimate by Johnson et al., an order of $N^{-3/2}$ accuracy can be obtained when continuous piecewise linear elements are used, where $N$ is the number of elements.

  • Keywords

Adaptive finite element, Nonlinear hyperbolic conservation law.

  • AMS Subject Headings

65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-29-491, author = {Zhengfu Xu, Jinchao Xu, and Chi-Wang Shu}, title = {A High Order Adaptive Finite Element Method for Solving Nonlinear Hyperbolic Conservation Laws}, journal = {Journal of Computational Mathematics}, year = {2011}, volume = {29}, number = {5}, pages = {491--500}, abstract = {

In this note, we apply the $h$-adaptive streamline diffusion finite element method with a small mesh-dependent artificial viscosity to solve nonlinear hyperbolic partial differential equations, with the objective of achieving high order accuracy and mesh efficiency. We compute the numerical solution to a steady state Burgers equation and the solution to a converging-diverging nozzle problem. The computational results verify that, by suitably choosing the artificial viscosity coefficient and applying the adaptive strategy based on a posterior error estimate by Johnson et al., an order of $N^{-3/2}$ accuracy can be obtained when continuous piecewise linear elements are used, where $N$ is the number of elements.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1105-m3392}, url = {http://global-sci.org/intro/article_detail/jcm/8490.html} }
TY - JOUR T1 - A High Order Adaptive Finite Element Method for Solving Nonlinear Hyperbolic Conservation Laws AU - Zhengfu Xu, Jinchao Xu, & Chi-Wang Shu JO - Journal of Computational Mathematics VL - 5 SP - 491 EP - 500 PY - 2011 DA - 2011/10 SN - 29 DO - http://doi.org/10.4208/jcm.1105-m3392 UR - https://global-sci.org/intro/article_detail/jcm/8490.html KW - Adaptive finite element, Nonlinear hyperbolic conservation law. AB -

In this note, we apply the $h$-adaptive streamline diffusion finite element method with a small mesh-dependent artificial viscosity to solve nonlinear hyperbolic partial differential equations, with the objective of achieving high order accuracy and mesh efficiency. We compute the numerical solution to a steady state Burgers equation and the solution to a converging-diverging nozzle problem. The computational results verify that, by suitably choosing the artificial viscosity coefficient and applying the adaptive strategy based on a posterior error estimate by Johnson et al., an order of $N^{-3/2}$ accuracy can be obtained when continuous piecewise linear elements are used, where $N$ is the number of elements.

Zhengfu Xu, Jinchao Xu, and Chi-Wang Shu. (2011). A High Order Adaptive Finite Element Method for Solving Nonlinear Hyperbolic Conservation Laws. Journal of Computational Mathematics. 29 (5). 491-500. doi:10.4208/jcm.1105-m3392
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