Volume 25, Issue 5
Reconstructed Discontinuous Galerkin Methods for Hyperbolic Diffusion Equations on Unstructured Grids

Jialin Lou, Xiaodong Liu, Hong Luo & Hiroaki Nishikawa

Commun. Comput. Phys., 25 (2019), pp. 1302-1327.

Published online: 2019-01

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  • Abstract

Reconstructed Discontinuous Galerkin (rDG) methods are presented for solving diffusion equations based on a first-order hyperbolic system (FOHS) formulation. The idea is to combine the advantages of the FOHS formulation and the rDG methods in an effort to develop a more reliable, accurate, efficient, and robust method for solving the diffusion equations. The developed hyperbolic rDG methods can be made to have higher-order accuracy than conventional DG methods with fewer degrees of freedom. A number of test cases for different diffusion equations are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods in comparison with the standard BR2 DG method. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous girds, and outperform the BR2 method in terms of the magnitude of the error, the order of accuracy, the size of time steps, and the CPU times required to achieve steady state solutions, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the diffusion equations.

  • Keywords

Discontinuous Galerkin methods, first-order hyperbolic system, unstructured grids.

  • AMS Subject Headings

76M10, 35L55

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-25-1302, author = {}, title = {Reconstructed Discontinuous Galerkin Methods for Hyperbolic Diffusion Equations on Unstructured Grids}, journal = {Communications in Computational Physics}, year = {2019}, volume = {25}, number = {5}, pages = {1302--1327}, abstract = {

Reconstructed Discontinuous Galerkin (rDG) methods are presented for solving diffusion equations based on a first-order hyperbolic system (FOHS) formulation. The idea is to combine the advantages of the FOHS formulation and the rDG methods in an effort to develop a more reliable, accurate, efficient, and robust method for solving the diffusion equations. The developed hyperbolic rDG methods can be made to have higher-order accuracy than conventional DG methods with fewer degrees of freedom. A number of test cases for different diffusion equations are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods in comparison with the standard BR2 DG method. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous girds, and outperform the BR2 method in terms of the magnitude of the error, the order of accuracy, the size of time steps, and the CPU times required to achieve steady state solutions, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the diffusion equations.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0186}, url = {http://global-sci.org/intro/article_detail/cicp/12952.html} }
TY - JOUR T1 - Reconstructed Discontinuous Galerkin Methods for Hyperbolic Diffusion Equations on Unstructured Grids JO - Communications in Computational Physics VL - 5 SP - 1302 EP - 1327 PY - 2019 DA - 2019/01 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2017-0186 UR - https://global-sci.org/intro/article_detail/cicp/12952.html KW - Discontinuous Galerkin methods, first-order hyperbolic system, unstructured grids. AB -

Reconstructed Discontinuous Galerkin (rDG) methods are presented for solving diffusion equations based on a first-order hyperbolic system (FOHS) formulation. The idea is to combine the advantages of the FOHS formulation and the rDG methods in an effort to develop a more reliable, accurate, efficient, and robust method for solving the diffusion equations. The developed hyperbolic rDG methods can be made to have higher-order accuracy than conventional DG methods with fewer degrees of freedom. A number of test cases for different diffusion equations are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods in comparison with the standard BR2 DG method. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous girds, and outperform the BR2 method in terms of the magnitude of the error, the order of accuracy, the size of time steps, and the CPU times required to achieve steady state solutions, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the diffusion equations.

Jialin Lou, Xiaodong Liu, Hong Luo & Hiroaki Nishikawa. (2020). Reconstructed Discontinuous Galerkin Methods for Hyperbolic Diffusion Equations on Unstructured Grids. Communications in Computational Physics. 25 (5). 1302-1327. doi:10.4208/cicp.OA-2017-0186
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