Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a new method called δ-ziti’s method for solving the governing equations. This method gives a new class of semi discrete, high-order scheme which are entropy conservative if the viscosity term is neglected. We implement a high resolution scheme for our mixed type problems that select the same viscosity solution as the Lax Friederich scheme with higher resolution. Several tests have been carried out to compare our results with those of [6] [9] [16], in the same situations, we obtained the same results but faster thanks to the CFL condition which reaches 0.8 and the simplicity of the method. We consider three types of pressure in these tests: Cubic, Van der Waals and linear in pieces. The comparison proved that the δ-ziti's method respects the generalized Liu entropy conditions, e.g. the existence of a viscous profile.
| Published in | Applied and Computational Mathematics (Volume 6, Issue 5) |
| DOI | 10.11648/j.acm.20170605.12 |
| Page(s) | 222-232 |
| Creative Commons |
This is an Open Access article, 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 or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2017. Published by Science Publishing Group |
Hyperbolic, Van Der Waals, δ-ziti’s Method, System Mixed Type, the Lax-Friedrichs Scheme, Shock Wave, Rarefaction Wave, Viscous Profile
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APA Style
Larbi Bsiss, Cherif Ziti. (2017). A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests. Applied and Computational Mathematics, 6(5), 222-232. https://doi.org/10.11648/j.acm.20170605.12
ACS Style
Larbi Bsiss; Cherif Ziti. A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests. Appl. Comput. Math. 2017, 6(5), 222-232. doi: 10.11648/j.acm.20170605.12
AMA Style
Larbi Bsiss, Cherif Ziti. A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests. Appl Comput Math. 2017;6(5):222-232. doi: 10.11648/j.acm.20170605.12
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Download@article{10.11648/j.acm.20170605.12,
author = {Larbi Bsiss and Cherif Ziti},
title = {A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests},
journal = {Applied and Computational Mathematics},
volume = {6},
number = {5},
pages = {222-232},
doi = {10.11648/j.acm.20170605.12},
url = {https://doi.org/10.11648/j.acm.20170605.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170605.12},
abstract = {Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a new method called δ-ziti’s method for solving the governing equations. This method gives a new class of semi discrete, high-order scheme which are entropy conservative if the viscosity term is neglected. We implement a high resolution scheme for our mixed type problems that select the same viscosity solution as the Lax Friederich scheme with higher resolution. Several tests have been carried out to compare our results with those of [6] [9] [16], in the same situations, we obtained the same results but faster thanks to the CFL condition which reaches 0.8 and the simplicity of the method. We consider three types of pressure in these tests: Cubic, Van der Waals and linear in pieces. The comparison proved that the δ-ziti's method respects the generalized Liu entropy conditions, e.g. the existence of a viscous profile.},
year = {2017}
}
TY - JOUR T1 - A New Entropic Riemann Solver of Conservation Law of Mixed Type Including Ziti’s δ-Method with some Experimental Tests AU - Larbi Bsiss AU - Cherif Ziti Y1 - 2017/09/26 PY - 2017 N1 - https://doi.org/10.11648/j.acm.20170605.12 DO - 10.11648/j.acm.20170605.12 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 222 EP - 232 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20170605.12 AB - Many problems in fluid mechanics and material sciences deal with liquid-vapour flows. In these flows, the ideal gas assumption is not accurate and the van der Waals equation of state is usually used. This equation of state is non-convex and causes the solution domain to have two hyperbolic regions separated by an elliptic region. Therefore, the governing equations of these flows have a mixed elliptic-hyperbolic nature. Numerical oscillations usually appear with standard finite-difference space discretization schemes, and they persist when the order of accuracy of the semi-discrete scheme is increased. In this study, we propose to use a new method called δ-ziti’s method for solving the governing equations. This method gives a new class of semi discrete, high-order scheme which are entropy conservative if the viscosity term is neglected. We implement a high resolution scheme for our mixed type problems that select the same viscosity solution as the Lax Friederich scheme with higher resolution. Several tests have been carried out to compare our results with those of [6] [9] [16], in the same situations, we obtained the same results but faster thanks to the CFL condition which reaches 0.8 and the simplicity of the method. We consider three types of pressure in these tests: Cubic, Van der Waals and linear in pieces. The comparison proved that the δ-ziti's method respects the generalized Liu entropy conditions, e.g. the existence of a viscous profile. VL - 6 IS - 5 ER -
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