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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.

  • AMS Subject Headings

65M08, 65M50, 76M12, 90C20

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COPYRIGHT: © Global Science Press

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@Article{CiCP-24-454, author = {Li Chen, Guanghui Hu and Ruo Li}, 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 AU - Li Chen, Guanghui Hu & Ruo Li 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 and Ruo Li. (2018). 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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