Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 18 (2023), 149 -- 162

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

THEORETICAL AND NUMERICAL RESULTS FOR THE NONLINEAR SHALLOW WATER PROBLEM

Mohammed Lachache and Fatma Zohra Nouri

Abstract. In this paper, the nonlinear shallow water problem is studied analytically and numerically. This problem has been studied by different authors in the linear case; here we consider a nonlinear one with non constant coefficients. We first start by a mathematical analysis, where the well posedness is proved via the use of traveling wave solutions. The existence and uniqueness of the solution is demonstrated by the means of Schaulder's and Banach's fixed point theorems. Then, we explore the proposed model and investigate the stability conditions and the efficiency numerically.

2020 Mathematics Subject Classification: 35A01; 35A02; 35C07; 35L55; 65N06
Keywords: Shallow water model; Traveling wave solutions; Well posedness; Finite differences.

Full text

References

  1. A. Assala, N. Djedaidi and F.Z. Nouri, Dynamical behaviour of miscibles fluids in porous media, Int J. Dynamical Systems and Differential Equations, Vol. 8(3) (2018), 176-189. Zbl 1442.76036. MR3825023.

  2. O. Besson, S. Kane, and M. Sy, On a 1D-Shallow Water Model: Existence of solution and numerical simulations. Revue Africaine de Recherche en Informatique et Mathématiques Appliquées 9 (2008), 247-260. MR2507505.

  3. D. Bresch, B. Desjardins, Sur un modéle de Saint-Venant visqueux et sa limite quasigéostrophique, C.R.Acad.Paris, Ser.I 335 (2002), 973-978. Zbl 1032.76012. MR1955592.

  4. R. Djemiat, B. Basti and N. Benhamidouche, Existence of traveling wave solutions for a free boundary problem of higher-order space-fractional wave equations, Appl. Math. E-Notes 22 (2022), 427-436. Zbl 1514.34014. MR4490773.

  5. F. Gerbeau, B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; Numerical results, Discrete and Continuous Dynamical Systems-series B. 1, No. 1 (2001), 89-102. Zbl 0997.76023. MR1821555.

  6. A. Granas, J. Dugundji, Fixed Point Theory; Springer: New York, NY, USA, (2003). Zbl 1025.47002. MR1987179.

  7. S.A. Kamboh, I.N. Sarbini, J. Labadin, M.O. Eze, Simulation of 2D Saint-Venant equations in open channel by using MATLAB, Journal of IT in Asia, vol 5(1) (2015), 15-22.

  8. F.Z. Nouri, N. Djedaidi and S. Gasmi, Interface dynamics for a bi-phasic problem in heterogeneous porous media, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 30, No. 1 (2023), 21-33. Zbl 1025.47002. MR4564219.



Mohammed Lachache
Mathematical Modeling and Numerical Simulation Research Laboratory, Badji Mokhtar University,
Annaba, Algeria.
E-mail: mohamed.lachache@univ-annaba.org, mohamed.lachache1996@gmail.com

Fatma Zohra Nouri
Mathematical Modeling and Numerical Simulation Research Laboratory, Badji Mokhtar University,
Annaba, Algeria.
E-mail: tassili.nan09@gmail.com

http://www.utgjiu.ro/math/sma