The article deals with the numerical simulations for equations of geophysical fluids. These physical phenomena are modeled by the Navier-Stokes equations which describe the motion of the fluid, the ocean currents, the flow of water in a pipe and many other fluid flow phenomenon.These equations are very useful because of there utility. The Navier-Stokes equations for incompressible flow are nonlinear partial differential equations that drive the motion of fluids in the approximation of continuous media. The existence of general solutions to the Navier Stokes equations have already proven but in this paper we have interested to the numerical solution of the incompressible Navier-Stokes equations. We get an optimal discretization of the Navier-Stokes equations and numerical approximations of the solution are also given. The convergence and the stabilityof the approximated system are proven. The numerical resolution is based on Hermite finite elements. The numerical system was expressed in matrix form for computation of velocity and the pression fieldsapproach using MATLAB software. Numerical results for velocity field in two dimensional space of the velocity u(x,y) and pression p(x,y,) are given. And finally we give physical interpretation of the results obtained.
Published in | International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 4) |
DOI | 10.11648/j.ijamtp.20210704.14 |
Page(s) | 112-125 |
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), 2021. Published by Science Publishing Group |
Numerical Simulation, Geophysical Models, Navier-Stokes Equation, Stokes Equations, Hermite Finite Elements, Numerical Simulations
[1] | S. Allgeyer, M. O. Bristeau, D. Froger, R. Hamouda, V. Jauzein, et al., Numerical approximation of the 3rd hydrostatic Navier-Stokes system with free surface, ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2019, 53 (6), 10.1051/m2an/2019044). (hal-01393147v5). |
[2] | R. Araya, G. R. Barrenechea, and F. Valentin, Stabilized finite element methods based on multiscale enrichment for the Stokes problem, SIAM J. Numer. Anal., Vol. 44, Number 1, pp. 322-348, 2006. |
[3] | R. Araya, A. H. Poza, and F. Valentin, An Adaptive Residual Local Projection Finite Element Method for the Navier-Stokes Equations, Advances in Computational Mathematics, Vol. 40, Number 5-6, pp. 1093-1119, 2014. |
[4] | H. Brezis, Analyse fonctionnelle: théorie et applications. Masson, Paris, 1983. |
[5] | F. Brezzi, J. Douglas and M. Fortin and D. Marini, Efficient rectangular mixed finite elements in two and three space variables, Mathematical Modelling and Numerical Analysis, 21 (4), (1987), p. 581-604. |
[6] | G. et F. Demengel Espaces fonctionnels: Utilisation dans la résolution des équations aux dérivées partielles. Paris, 2007, EDP Sciences. |
[7] | A. M. Doglioli, Modélisation de la circulation océanique, Notes de cours et de travaux dirigés, Marseille, France, Avril 2015. |
[8] | H. Le Dret, Equations aux dérivées partielles elliptiques non linéaires, Université Pierre et Marie Curie, Paris, Springer, 2013. |
[9] | A. Ern, Aide mémoire Eléments finis. Dunod, Paris, 2013. |
[10] | A. Fortin and A. Garon, Les ééments finis: de la théorie à la pratique, Montreal, 2007- 2011. |
[11] | R. Glowinski, Numerical Methods for Nonlinear Variational problem, Library of Congress Control Number: 2007942575, 2008, 1984 Springer-Verlag Berlin Heidelberg. |
[12] | R. Glowinski Lectures on Numerical MethodS for Non-Linear Variational Problems, Tata Institute of Fundamental Research Bombay, Springer- Verlag, 1980. |
[13] | R. B. Hamouda, Notions de Mécaniques des Fluides, cours et exercices, Centre de Publication Universitaire, Octobre 2008. |
[14] | T. Kato and H. Fujita, On the nonstationary Navier- Stokes system, Rendiconti del Seminario Matematico della Universit di Padova, tome 32 (1962), p. 243-260. |
[15] | T. Kheladi, Equations de Navier-Stokes, Mémoire de Master, Université Abderrahmane Mira-Béjaia, juin 2013. |
[16] | J. Leray Sur le mouvement d’un fluide visqueux emplissant l’espace, Compte rendu de l’Académie des des sciences, février 1933, T. 196, p. 527. |
[17] | J. L. Lions, Quelques résultats d’existence dans les équations aux dérivées partielles non linéaires, Bulletin de la S. M. F, tome 87 (1959), pp. 245-273. |
[18] | U. Lorraine, Analyse numérique des équations de Navier- Stokes, Septembre 2004. |
[19] | H. Oudin, Méthode des éléments finis, Centrale Nantes, 2003. |
[20] | M. Postel, Introduction au logiciel Matlab, Septembre 2004. |
[21] | A. M. Quarteroni, R. Sacco and F. Saleri, Méthodes numériques, Algorithmes, analyse et applications, Springer-Verlag, 2006. |
[22] | R. Temam, Navier-Stokes Equation, North-Holland Publishing Compagny, New York-Oxford, 1979. |
[23] | R. Temam, Navier-Stokes equations, Editors: J. L. Lions, Paris; G. Papanicolaou, New York; T. T. Rockafeller, Seattle, North-Holland Publishing Company, first edition 1977, revised edition 1949. |
APA Style
Ibrahima Thiam, Babou Khady Thiam, Ibrahima Faye. (2021). Numerical Simulation of Ocean Currents with Hermite IFinite Elements. International Journal of Applied Mathematics and Theoretical Physics, 7(4), 112-125. https://doi.org/10.11648/j.ijamtp.20210704.14
ACS Style
Ibrahima Thiam; Babou Khady Thiam; Ibrahima Faye. Numerical Simulation of Ocean Currents with Hermite IFinite Elements. Int. J. Appl. Math. Theor. Phys. 2021, 7(4), 112-125. doi: 10.11648/j.ijamtp.20210704.14
AMA Style
Ibrahima Thiam, Babou Khady Thiam, Ibrahima Faye. Numerical Simulation of Ocean Currents with Hermite IFinite Elements. Int J Appl Math Theor Phys. 2021;7(4):112-125. doi: 10.11648/j.ijamtp.20210704.14
@article{10.11648/j.ijamtp.20210704.14, author = {Ibrahima Thiam and Babou Khady Thiam and Ibrahima Faye}, title = {Numerical Simulation of Ocean Currents with Hermite IFinite Elements}, journal = {International Journal of Applied Mathematics and Theoretical Physics}, volume = {7}, number = {4}, pages = {112-125}, doi = {10.11648/j.ijamtp.20210704.14}, url = {https://doi.org/10.11648/j.ijamtp.20210704.14}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210704.14}, abstract = {The article deals with the numerical simulations for equations of geophysical fluids. These physical phenomena are modeled by the Navier-Stokes equations which describe the motion of the fluid, the ocean currents, the flow of water in a pipe and many other fluid flow phenomenon.These equations are very useful because of there utility. The Navier-Stokes equations for incompressible flow are nonlinear partial differential equations that drive the motion of fluids in the approximation of continuous media. The existence of general solutions to the Navier Stokes equations have already proven but in this paper we have interested to the numerical solution of the incompressible Navier-Stokes equations. We get an optimal discretization of the Navier-Stokes equations and numerical approximations of the solution are also given. The convergence and the stabilityof the approximated system are proven. The numerical resolution is based on Hermite finite elements. The numerical system was expressed in matrix form for computation of velocity and the pression fieldsapproach using MATLAB software. Numerical results for velocity field in two dimensional space of the velocity u(x,y) and pression p(x,y,) are given. And finally we give physical interpretation of the results obtained.}, year = {2021} }
TY - JOUR T1 - Numerical Simulation of Ocean Currents with Hermite IFinite Elements AU - Ibrahima Thiam AU - Babou Khady Thiam AU - Ibrahima Faye Y1 - 2021/12/24 PY - 2021 N1 - https://doi.org/10.11648/j.ijamtp.20210704.14 DO - 10.11648/j.ijamtp.20210704.14 T2 - International Journal of Applied Mathematics and Theoretical Physics JF - International Journal of Applied Mathematics and Theoretical Physics JO - International Journal of Applied Mathematics and Theoretical Physics SP - 112 EP - 125 PB - Science Publishing Group SN - 2575-5927 UR - https://doi.org/10.11648/j.ijamtp.20210704.14 AB - The article deals with the numerical simulations for equations of geophysical fluids. These physical phenomena are modeled by the Navier-Stokes equations which describe the motion of the fluid, the ocean currents, the flow of water in a pipe and many other fluid flow phenomenon.These equations are very useful because of there utility. The Navier-Stokes equations for incompressible flow are nonlinear partial differential equations that drive the motion of fluids in the approximation of continuous media. The existence of general solutions to the Navier Stokes equations have already proven but in this paper we have interested to the numerical solution of the incompressible Navier-Stokes equations. We get an optimal discretization of the Navier-Stokes equations and numerical approximations of the solution are also given. The convergence and the stabilityof the approximated system are proven. The numerical resolution is based on Hermite finite elements. The numerical system was expressed in matrix form for computation of velocity and the pression fieldsapproach using MATLAB software. Numerical results for velocity field in two dimensional space of the velocity u(x,y) and pression p(x,y,) are given. And finally we give physical interpretation of the results obtained. VL - 7 IS - 4 ER -