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Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables
We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. T...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Springer International Publishing
2016
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6512000/ https://www.ncbi.nlm.nih.gov/pubmed/31149558 http://dx.doi.org/10.1186/s40668-015-0014-x |
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author | Zanotti, Olindo Dumbser, Michael |
author_facet | Zanotti, Olindo Dumbser, Michael |
author_sort | Zanotti, Olindo |
collection | PubMed |
description | We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. To obtain a conservative method, the underlying finite volume scheme is still written in terms of the cell averages of the conserved quantities. Therefore, our new approach performs the spatial WENO reconstruction twice: the first WENO reconstruction is carried out on the known cell averages of the conservative variables. The WENO polynomials are then used at the cell centers to compute point values of the conserved variables, which are subsequently converted into point values of the primitive variables. This is the only place where the conversion from conservative to primitive variables is needed in the new scheme. Then, a second WENO reconstruction is performed on the point values of the primitive variables to obtain piecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials are subsequently evolved in time with a novel space-time finite element predictor that is directly applied to the governing PDE written in primitive form. The resulting space-time polynomials of the primitive variables can then be directly used as input for the numerical fluxes at the cell boundaries in the underlying conservative finite volume scheme. Hence, the number of necessary conversions from the conserved to the primitive variables is reduced to just one single conversion at each cell center. We have verified the validity of the new approach over a wide range of hyperbolic systems, including the classical Euler equations of gas dynamics, the special relativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD) equations, as well as the Baer-Nunziato model for compressible two-phase flows. In all cases we have noticed that the new ADER schemes provide less oscillatory solutions when compared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for the RMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the overall accuracy is improved and the CPU time is reduced by about 25 %. Because of its increased accuracy and due to the reduced computational cost, we recommend to use this version of ADER as the standard one in the relativistic framework. At the end of the paper, the new approach has also been extended to ADER-DG schemes on space-time adaptive grids (AMR). |
format | Online Article Text |
id | pubmed-6512000 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2016 |
publisher | Springer International Publishing |
record_format | MEDLINE/PubMed |
spelling | pubmed-65120002019-05-28 Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables Zanotti, Olindo Dumbser, Michael Comput Astrophys Cosmol Research We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. To obtain a conservative method, the underlying finite volume scheme is still written in terms of the cell averages of the conserved quantities. Therefore, our new approach performs the spatial WENO reconstruction twice: the first WENO reconstruction is carried out on the known cell averages of the conservative variables. The WENO polynomials are then used at the cell centers to compute point values of the conserved variables, which are subsequently converted into point values of the primitive variables. This is the only place where the conversion from conservative to primitive variables is needed in the new scheme. Then, a second WENO reconstruction is performed on the point values of the primitive variables to obtain piecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials are subsequently evolved in time with a novel space-time finite element predictor that is directly applied to the governing PDE written in primitive form. The resulting space-time polynomials of the primitive variables can then be directly used as input for the numerical fluxes at the cell boundaries in the underlying conservative finite volume scheme. Hence, the number of necessary conversions from the conserved to the primitive variables is reduced to just one single conversion at each cell center. We have verified the validity of the new approach over a wide range of hyperbolic systems, including the classical Euler equations of gas dynamics, the special relativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD) equations, as well as the Baer-Nunziato model for compressible two-phase flows. In all cases we have noticed that the new ADER schemes provide less oscillatory solutions when compared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for the RMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the overall accuracy is improved and the CPU time is reduced by about 25 %. Because of its increased accuracy and due to the reduced computational cost, we recommend to use this version of ADER as the standard one in the relativistic framework. At the end of the paper, the new approach has also been extended to ADER-DG schemes on space-time adaptive grids (AMR). Springer International Publishing 2016-01-13 2016 /pmc/articles/PMC6512000/ /pubmed/31149558 http://dx.doi.org/10.1186/s40668-015-0014-x Text en © Zanotti and Dumbser 2016 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
spellingShingle | Research Zanotti, Olindo Dumbser, Michael Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables |
title | Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables |
title_full | Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables |
title_fullStr | Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables |
title_full_unstemmed | Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables |
title_short | Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables |
title_sort | efficient conservative ader schemes based on weno reconstruction and space-time predictor in primitive variables |
topic | Research |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6512000/ https://www.ncbi.nlm.nih.gov/pubmed/31149558 http://dx.doi.org/10.1186/s40668-015-0014-x |
work_keys_str_mv | AT zanottiolindo efficientconservativeaderschemesbasedonwenoreconstructionandspacetimepredictorinprimitivevariables AT dumbsermichael efficientconservativeaderschemesbasedonwenoreconstructionandspacetimepredictorinprimitivevariables |