Finite-volume two-step scheme for solving the shear shallow water model

H. S. Alayachi; Mahmoud A. E. Abdelrahman; Kamel Mohamed; H. S. Alayachi; Mahmoud A. E. Abdelrahman; Kamel Mohamed

doi:10.3934/math.2024980

AIMS Mathematics

2024, Volume 9, Issue 8: 20118-20135. doi: 10.3934/math.2024980

Previous Article Next Article

Research article Special Issues

Finite-volume two-step scheme for solving the shear shallow water model

1.
Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
3.
Department of Mathematics, Faculty of Science, New Valley University, New Valley, Egypt

Received: 30 January 2024 Revised: 12 May 2024 Accepted: 29 May 2024 Published: 20 June 2024
MSC : 35L03, 35L65, 65M08

The shear shallow water (SSW) model introduces an approximation for shallow water flows by including the effect of vertical shear in the system. Six non-linear hyperbolic partial differential equations with non-conservative laws make up this system. Shear, contact, rarefaction, and shock waves are all admissible in this model. We developed the finite-volume two-step scheme, the so-called generalized Rusanov (G. Rusanov) scheme, for solving the SSW model. This method is split into two stages. The first one relies on a local parameter that permits control over the diffusion. In stage two, the conservation equation is recovered. Numerous numerical instances were taken into consideration. We clarified that the G. Rusanov scheme satisfied the C-property. We also compared the numerical solutions with those obtained from the Rusanov, Lax-Friedrichs, and reference solutions. Finally, the G. Rusanov technique may be applied for solving a wide range of additional models in developed physics and applied science.
- finite-volume two-step scheme,
- shear shallow water model,
- nonlinear elementary waves
Citation: H. S. Alayachi, Mahmoud A. E. Abdelrahman, Kamel Mohamed. Finite-volume two-step scheme for solving the shear shallow water model[J]. AIMS Mathematics, 2024, 9(8): 20118-20135. doi: 10.3934/math.2024980

Related Papers:

Abstract

The shear shallow water (SSW) model introduces an approximation for shallow water flows by including the effect of vertical shear in the system. Six non-linear hyperbolic partial differential equations with non-conservative laws make up this system. Shear, contact, rarefaction, and shock waves are all admissible in this model. We developed the finite-volume two-step scheme, the so-called generalized Rusanov (G. Rusanov) scheme, for solving the SSW model. This method is split into two stages. The first one relies on a local parameter that permits control over the diffusion. In stage two, the conservation equation is recovered. Numerous numerical instances were taken into consideration. We clarified that the G. Rusanov scheme satisfied the C-property. We also compared the numerical solutions with those obtained from the Rusanov, Lax-Friedrichs, and reference solutions. Finally, the G. Rusanov technique may be applied for solving a wide range of additional models in developed physics and applied science.

References

[1]	G. B. Whitham, Linear and nonlinear waves, Hoboken: John Wiley & Sons, 1999. http://doi.org/10.1002/9781118032954
[2]	M. A. E. Abdelrahman, On the shallow water equations, ZNA, 72 (2017), 873–879. https://doi.org/10.1515/zna-2017-0146 doi: 10.1515/zna-2017-0146
[3]	K. Mohamed, S. Sahmim, F. Benkhaldoun, M. A. E. Abdelrahman, Some recent finite volume schemes for one and two layers shallow water equations with variable density, Math. Methods Appl. Sci., 46 (2023), 12979–12995. https://doi.org/10.1002/mma.9227 doi: 10.1002/mma.9227
[4]	K. Mohamed, H. S. Alayachi, M. A. E. Abdelrahman, The mR scheme to the shallow water equation with horizontal density gradients in one and two dimensions, AIMS Mathematics, 8 (2023), 25754–25771. http://doi.org/10.3934/math.20231314 doi: 10.3934/math.20231314
[5]	J. Ren, O. A. Ilhan, H. Bulut, J. Manafian, Multiple rogue wave, dark, bright, and solitary wave solutions to the KP–BBM equation, J. Geom. Phys., 164 (2021), 104159. https://doi.org/10.1016/j.geomphys.2021.104159 doi: 10.1016/j.geomphys.2021.104159
[6]	X. Zhou, O. A. Ilhan, J. Manafian, G. Singh, N. S. Tuguz, N-lump and interaction solutions of localized waves to the (2+1)-dimensional generalized KDKK equation, J. Geom. Phys., 168 (2021), 104312. https://doi.org/10.1016/j.geomphys.2021.104312 doi: 10.1016/j.geomphys.2021.104312
[7]	X. Hong, J. Manafian, O. A. Ilhan, A. I. A. Alkireet, M. K. M. Nasution, Multiple soliton solutions of the generalized Hirota-Satsuma-Ito equation arising in shallow water wave, J. Geom. Phys., 170 (2021), 104338. https://doi.org/10.1016/j.geomphys.2021.104338 doi: 10.1016/j.geomphys.2021.104338
[8]	W. Cai, R. Mohammaditab, G. Fathi, K. Wakil, A. G. Ebadi, N. Ghadimi, Optimal bidding and offering strategies of compressed air energy storage: A hybrid robust-stochastic approach, Renewable Energy, 143 (2019), 1–8. https://doi.org/10.1016/j.renene.2019.05.008 doi: 10.1016/j.renene.2019.05.008
[9]	V. M. Teshukov, Gas-dynamic analogy for vortex free-boundary flows, J. Appl. Mech. Tech. Phys., 48 (2007), 303–309. https://doi.org/10.1007/s10808-007-0039-2 doi: 10.1007/s10808-007-0039-2
[10]	M. M. A. Khater, S. H. Alfalqi, J. F. Alzaidi, S. A. Salama, F. Wang, Plenty of wave solutions to the ill-posed Boussinesq dynamic wave equation under shallow water beneath gravity, AIMS Mathematics, 7 (2022), 54–81. http://doi.org/10.3934/math.2022004 doi: 10.3934/math.2022004
[11]	A. I. Delis, H. Guillard, Y. C. Tai, Numerical simulations of hydraulic jumps with the shear shallow water model, SMAI J. Comput. Math., 4 (2018), 319–344. http://doi.org/10.5802/smai-jcm.37 doi: 10.5802/smai-jcm.37
[12]	J. Zhang, F. Wang, S. Nadeem, M. Sun, Simulation of linear and nonlinear advection-diffusion problems by the direct radial basis function collocation method, Int. Commun. Heat Mass Trans., 130 (2022), 105775. https://doi.org/10.1016/j.icheatmasstransfer.2021.105775 doi: 10.1016/j.icheatmasstransfer.2021.105775
[13]	F. Wang, E. Hou, I. Ahmad, H. Ahmad, Y. Gu, An efficient meshless method for hyperbolic telegraph equations in (1+1) dimensions, Comput. Model. Eng. Sci., 128 (2021), 687–698. https://doi.10.32604/cmes.2021.014739 doi: 10.32604/cmes.2021.014739
[14]	Z. Zhang, F. Wang, J. Zhang, The space-time meshless methods for the solution of one-dimensional Klein-Gordon equations, Wuhan Univ. J. Nat. Sci., 27 (2022), 313–320. https://doi.10.1051/wujns/2022274313 doi: 10.1051/wujns/2022274313
[15]	O. A. Ilhan, J. Manafian, M. Shahriari, Lump wave solutions and the interaction phenomenon for a variable-coefficient Kadomtsev–Petviashvili equation, Comput. Math. Appl., 78 (2019), 2429–2448. https://doi.org/10.1016/j.camwa.2019.03.048 doi: 10.1016/j.camwa.2019.03.048
[16]	H. Zhang, J. Manafian, G. Singh, O. A. Ilhan, A. O. Zekiy, N-lump and interaction solutions of localized waves to the (2+1)-dimensional generalized KP equation, Results Phys., 25 (2021), 104168. https://doi.org/10.1016/j.rinp.2021.104168 doi: 10.1016/j.rinp.2021.104168
[17]	Y. Gu, S. Malmir, J. Manafian, O. A. Ilhan, A. Alizadeh, A. J. Othman, Variety interaction between K-lump and K-kink solutions for the (3+1)-D Burger system by bilinear analysis, Results Phys., 43 (2022), 131–142. https://doi.org/10.1016/j.rinp.2021.104490 doi: 10.1016/j.rinp.2021.104490
[18]	A. Yadav, D. Bhoriya, H. Kumar, P. Chandrashekar, Entropy stable schemes for the shear shallow water model equations, J. Sci. Comput., 97 (2023), 131–142. https://doi.org/10.1007/s10915-023-02374-4 doi: 10.1007/s10915-023-02374-4
[19]	K. Mohamed, Simulation numérique en volume finis, de problémes d'écoulements multidimensionnels raides, par un schéma de flux á deux pas, University of Paris, PhD thesis, 2005.
[20]	K. Mohamed, M. Seaid, M. Zahri, A finite volume method for scalar conservation laws with stochastic time-space dependent flux function, J. Comput. Appl. Math., 237 (2013), 614–632. https://doi.org/10.1016/j.cam.2012.07.014 doi: 10.1016/j.cam.2012.07.014
[21]	K. Mohamed, S. Sahmim, M. A. E. Abdelrahman, A predictor-corrector scheme for simulation of two-phase granular flows over a moved bed with a variable topography, Eur. J. Mech. - B/Fluids, 96 (2022), 39–50. https://doi.org/10.1016/j.euromechflu.2022.07.001 doi: 10.1016/j.euromechflu.2022.07.001
[22]	K. Mohamed, F. Benkhaldoun, A modified Rusanov scheme for shallow water equations with topography and two phase flows, Eur. Phys. J. Plus, 131 (2016), 207. https://doi.org/10.1140/epjp/i2016-16207-3 doi: 10.1140/epjp/i2016-16207-3
[23]	K. Mohamed, A finite volume method for numerical simulation of shallow water models with porosity, Comput. Fluids, 104 (2014), 9–19. https://doi.org/10.1016/j.compfluid.2014.07.020 doi: 10.1016/j.compfluid.2014.07.020
[24]	B. Nkonga1, P. Chandrashekar, Exact solution for Riemann problems of the shear shallow water model, ESAIM: Math. Modell. Numer. Anal., 56 (2022), 1115–1150.
[25]	F. Benkhaldoun, K. Mohamed, M. Seaid, A Generalized Rusanov method for Saint-Venant Equations with Variable Horizontal Density, Finite Volumes for Complex Applications VI Problems & Perspectives, 2011, 89–96.
[26]	K. Mohamed, A. R. Seadawy, Finite volume scheme for numerical simulation of the sediment transport model, Int. J. Modern Phys. B, 33 (2019), 1950283. https://doi.org/10.1142/S0217979219502837 doi: 10.1142/S0217979219502837
[27]	K. Mohamed, M. A. E. Abdelrahman, The modified Rusanov scheme for solving the ultra-relativistic Euler equations, Eur. J. Mech. - B/Fluids, 90 (2021), 89–98. https://doi.org/10.1016/j.euromechflu.2021.07.014 doi: 10.1016/j.euromechflu.2021.07.014
[28]	R. J. LeVeque, Numerical methods for conservation laws, Basel: Birkhäuser Verlag, 1992.
[29]	B. van Leer, Towards the ultimate conservative difference schemes. V. A second-order Ssequal to Godunov's method, J. Comput. Phys., 32 (1979), 101–136. https://doi.org/10.1016/0021-9991(79)90145-1 doi: 10.1016/0021-9991(79)90145-1
[30]	A. Bermudez, M. E. Vazquez, Upwind methods for hyperbolic conservation laws with source term, Comput. Fluids, 23 (1994), 1049–1071. https://doi.org/10.1016/0045-7930(94)90004-3 doi: 10.1016/0045-7930(94)90004-3
[31]	L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms, Math. Models Methods Appl. Sci., 11 (2001), 339–365. http://doi.org/10.1142/S021820250100088X doi: 10.1142/S021820250100088X
[32]	P. Chandrashekar, B. Nkonga, A. Kumari Meena, A. Bhole, A path conservative ﬁnite volume method for a shear shallow water model, J. Comput. Phys., 413 (2020), 109457. https://doi.org/10.1016/j.jcp.2020.109457 doi: 10.1016/j.jcp.2020.109457

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)