Volume 24, Issue 2
Integrated Linear Reconstruction for Finite Volume Scheme on Arbitrary Unstructured Grids

Li Chen, Guanghui Hu & Ruo Li

Commun. Comput. Phys., 24 (2018), pp. 454-480.

Published online: 2018-08

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  • Abstract

In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp. 1172–1197, (2016)], an integrated linear reconstruction was proposed for finite volume methods on unstructured grids. However, the geometric hypothesis of the mesh to enforce a local maximum principle is too restrictive to be satisfied by, for example, locally refined meshes or distorted meshes generated by arbitrary Lagrangian-Eulerian methods in practical applications. In this paper, we propose an improved integrated linear reconstruction approach to get rid of the geometric hypothesis. The resulting optimization problem is a convex quadratic programming problem, and hence can be solved efficiently by classical active-set methods. The features of the improved integrated linear reconstruction include that i). the local maximum principle is fulfilled on arbitrary unstructured grids, ii). the reconstruction is parameter-free, and iii). the finite volume scheme is positivity-preserving when the reconstruction is generalized to the Euler equations. A variety of numerical experiments are presented to demonstrate the performance of this method.

  • Keywords

Linear reconstruction, finite volume method, local maximum principle, positivity-preserving, quadratic programming.

  • AMS Subject Headings

65M08, 65M50, 76M12, 90C20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-454, author = {}, title = {Integrated Linear Reconstruction for Finite Volume Scheme on Arbitrary Unstructured Grids}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {2}, pages = {454--480}, abstract = {

In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp. 1172–1197, (2016)], an integrated linear reconstruction was proposed for finite volume methods on unstructured grids. However, the geometric hypothesis of the mesh to enforce a local maximum principle is too restrictive to be satisfied by, for example, locally refined meshes or distorted meshes generated by arbitrary Lagrangian-Eulerian methods in practical applications. In this paper, we propose an improved integrated linear reconstruction approach to get rid of the geometric hypothesis. The resulting optimization problem is a convex quadratic programming problem, and hence can be solved efficiently by classical active-set methods. The features of the improved integrated linear reconstruction include that i). the local maximum principle is fulfilled on arbitrary unstructured grids, ii). the reconstruction is parameter-free, and iii). the finite volume scheme is positivity-preserving when the reconstruction is generalized to the Euler equations. A variety of numerical experiments are presented to demonstrate the performance of this method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0137}, url = {http://global-sci.org/intro/article_detail/cicp/12248.html} }
TY - JOUR T1 - Integrated Linear Reconstruction for Finite Volume Scheme on Arbitrary Unstructured Grids JO - Communications in Computational Physics VL - 2 SP - 454 EP - 480 PY - 2018 DA - 2018/08 SN - 24 DO - http://doi.org/10.4208/cicp.OA-2017-0137 UR - https://global-sci.org/intro/article_detail/cicp/12248.html KW - Linear reconstruction, finite volume method, local maximum principle, positivity-preserving, quadratic programming. AB -

In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp. 1172–1197, (2016)], an integrated linear reconstruction was proposed for finite volume methods on unstructured grids. However, the geometric hypothesis of the mesh to enforce a local maximum principle is too restrictive to be satisfied by, for example, locally refined meshes or distorted meshes generated by arbitrary Lagrangian-Eulerian methods in practical applications. In this paper, we propose an improved integrated linear reconstruction approach to get rid of the geometric hypothesis. The resulting optimization problem is a convex quadratic programming problem, and hence can be solved efficiently by classical active-set methods. The features of the improved integrated linear reconstruction include that i). the local maximum principle is fulfilled on arbitrary unstructured grids, ii). the reconstruction is parameter-free, and iii). the finite volume scheme is positivity-preserving when the reconstruction is generalized to the Euler equations. A variety of numerical experiments are presented to demonstrate the performance of this method.

Li Chen, Guanghui Hu & Ruo Li. (2020). Integrated Linear Reconstruction for Finite Volume Scheme on Arbitrary Unstructured Grids. Communications in Computational Physics. 24 (2). 454-480. doi:10.4208/cicp.OA-2017-0137
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