Volume 29, Issue 4
Investigation of Riemann Solver with Internal Reconstruction (RSIR) for the Euler Equations

Alexandre Chiapolino, Richard SaurelEleuterio Toro

Commun. Comput. Phys., 29 (2021), pp. 1059-1094.

Published online: 2021-02

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  • Abstract

The Riemann solver with internal reconstruction (RSIR) of Carmouze et al. (2020) is investigated, revisited and improved for the Euler equations. In this reference, the RSIR approach has been developed to address the numerical resolution of the non-equilibrium two-phase flow model of Saurel et al. (2017). The main idea is to reconstruct two intermediate states from the knowledge of a simple and robust intercell state such as HLL, regardless the number of waves present in the Riemann problem. Such reconstruction improves significantly the accuracy of the HLL solution, preserves robustness and maintains stationary discontinuities. Consequently, when dealing with complex flow models, such as the aforementioned one, RSIR-type solvers are quite easy to derive compared to HLLC-type ones that may require a tedious analysis of the governing equations across the different waves. In the present contribution, the RSIR solver of Carmouze et al. (2020) is investigated, revisited and improved with the help of thermodynamic considerations, making a simple, accurate, robust and positive Riemann solver. It is also demonstrated that the RSIR solver is strictly equivalent to the HLLC solver of Toro et al. (1994) for the Euler equations when the Rankine-Hugoniot relations are used. In that sense, the RSIR approach recovers the HLLC solver in some limit as well as the HLL one in another limit and can be simplified at different levels when complex systems of equations are addressed. For the sake of clarity and simplicity, the derivations are performed in the context of the one-dimensional Euler equations. Comparisons and validations against the conventional HLLC solver and exact solutions are presented.

  • Keywords

Riemann solver, RSIR, HLL, HLLC, hyperbolic.

  • AMS Subject Headings

35Q31, 35Q35, 76-11, 76L05, 76M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-1059, author = {Chiapolino , Alexandre and Saurel , Richard and Toro , Eleuterio}, title = {Investigation of Riemann Solver with Internal Reconstruction (RSIR) for the Euler Equations}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {4}, pages = {1059--1094}, abstract = {

The Riemann solver with internal reconstruction (RSIR) of Carmouze et al. (2020) is investigated, revisited and improved for the Euler equations. In this reference, the RSIR approach has been developed to address the numerical resolution of the non-equilibrium two-phase flow model of Saurel et al. (2017). The main idea is to reconstruct two intermediate states from the knowledge of a simple and robust intercell state such as HLL, regardless the number of waves present in the Riemann problem. Such reconstruction improves significantly the accuracy of the HLL solution, preserves robustness and maintains stationary discontinuities. Consequently, when dealing with complex flow models, such as the aforementioned one, RSIR-type solvers are quite easy to derive compared to HLLC-type ones that may require a tedious analysis of the governing equations across the different waves. In the present contribution, the RSIR solver of Carmouze et al. (2020) is investigated, revisited and improved with the help of thermodynamic considerations, making a simple, accurate, robust and positive Riemann solver. It is also demonstrated that the RSIR solver is strictly equivalent to the HLLC solver of Toro et al. (1994) for the Euler equations when the Rankine-Hugoniot relations are used. In that sense, the RSIR approach recovers the HLLC solver in some limit as well as the HLL one in another limit and can be simplified at different levels when complex systems of equations are addressed. For the sake of clarity and simplicity, the derivations are performed in the context of the one-dimensional Euler equations. Comparisons and validations against the conventional HLLC solver and exact solutions are presented.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0083}, url = {http://global-sci.org/intro/article_detail/cicp/18646.html} }
TY - JOUR T1 - Investigation of Riemann Solver with Internal Reconstruction (RSIR) for the Euler Equations AU - Chiapolino , Alexandre AU - Saurel , Richard AU - Toro , Eleuterio JO - Communications in Computational Physics VL - 4 SP - 1059 EP - 1094 PY - 2021 DA - 2021/02 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0083 UR - https://global-sci.org/intro/article_detail/cicp/18646.html KW - Riemann solver, RSIR, HLL, HLLC, hyperbolic. AB -

The Riemann solver with internal reconstruction (RSIR) of Carmouze et al. (2020) is investigated, revisited and improved for the Euler equations. In this reference, the RSIR approach has been developed to address the numerical resolution of the non-equilibrium two-phase flow model of Saurel et al. (2017). The main idea is to reconstruct two intermediate states from the knowledge of a simple and robust intercell state such as HLL, regardless the number of waves present in the Riemann problem. Such reconstruction improves significantly the accuracy of the HLL solution, preserves robustness and maintains stationary discontinuities. Consequently, when dealing with complex flow models, such as the aforementioned one, RSIR-type solvers are quite easy to derive compared to HLLC-type ones that may require a tedious analysis of the governing equations across the different waves. In the present contribution, the RSIR solver of Carmouze et al. (2020) is investigated, revisited and improved with the help of thermodynamic considerations, making a simple, accurate, robust and positive Riemann solver. It is also demonstrated that the RSIR solver is strictly equivalent to the HLLC solver of Toro et al. (1994) for the Euler equations when the Rankine-Hugoniot relations are used. In that sense, the RSIR approach recovers the HLLC solver in some limit as well as the HLL one in another limit and can be simplified at different levels when complex systems of equations are addressed. For the sake of clarity and simplicity, the derivations are performed in the context of the one-dimensional Euler equations. Comparisons and validations against the conventional HLLC solver and exact solutions are presented.

Alexandre Chiapolino, Richard Saurel & Eleuterio Toro. (2021). Investigation of Riemann Solver with Internal Reconstruction (RSIR) for the Euler Equations. Communications in Computational Physics. 29 (4). 1059-1094. doi:10.4208/cicp.OA-2020-0083
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