Lie Symmetry Analysis of the One-Dimensional Saint-Venant-Exner Model
Abstract
:1. Introduction
2. Lie Symmetries for the Saint-Venant-Exner Model
3. One-Dimensional Optimal System
4. Similarity Transformations
4.1. Reduction with
4.2. Reduction with
4.3. Reduction with
4.4. Reduction with
4.5. Reduction with Solution
4.6. Reduction with
4.7. Reduction with
4.8. Reduction with
4.9. Reduction with
4.10. Reduction with
4.11. Reduction with
4.12. Reduction with
5. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Caleffi, V.; Valiani, A.; Zanni, A. Finite volume method for simulating extreme flood events in natural channels. J. Hydraul. Res. 2003, 41, 167. [Google Scholar] [CrossRef]
- Akkermans, R.A.D.; Kamp, L.P.J.; Clercx, H.J.H.; van Heijst, G.J.F. Three-Dimensional flow in electromagnetically driven shallow two-layer fluids. Phys. Rev. E 2010, 82, 026314. [Google Scholar] [CrossRef] [PubMed]
- Vallis, G.K. Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Jung, J.; Hwang, J.H.; Borthwick, A.G.L. Piston-Driven Numerical Wave Tank Based on WENO Solver of Well-Balanced Shallow Water Equations. KSCE J. Civ. Eng. 2020, 24, 1959. [Google Scholar] [CrossRef]
- Kurganov, A.; Liu, Y.L.; Zeitlin, V. Moist-Convective thermal rotating shallow water model. Phys. Fluids 2020, 32, 7757. [Google Scholar] [CrossRef]
- Zhu, M.; Wang, Y. Wave-Breaking phenomena for a weakly dissipative shallow water equation. Z. Angew. Phys. 2020, 71, 96. [Google Scholar] [CrossRef]
- Khalique, C.M.; Plaatjie, K. Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation. Mathematics 2021, 9, 1439. [Google Scholar] [CrossRef]
- Bagchi, B.; Das, S.; Ganguly, A. New exact solutions of a generalized shallow water wave equation. Phys. Scr. 2010, 82, 025002. [Google Scholar] [CrossRef]
- Lai, S.; Wang, A. The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation. Abstr. Appl. Anal. 2021, 11, 1. [Google Scholar] [CrossRef]
- Zeidan, D.; Romenski, E.; Slaouti, A.; Toro, E.F. Numerical Solution for Hyperbolic Conservative two-phase flow equations. Int. J. Num. Meth. Fluids 2007, 54, 393. [Google Scholar] [CrossRef]
- Zhai, J.; Liu, W.; Yuan, L. Solving two-phase shallow granular flow equations with a well-balanced NOC scheme on multiple GPUs. Comput. Fluids 2016, 134, 90. [Google Scholar] [CrossRef]
- Stoker, J. Water Waves: The Mathematical Theory with Applications; Willey: Hoboken, NJ, USA, 1992. [Google Scholar]
- Whitham, G.B. Linear and Non-linear Waves; Willey: New York, NY, USA, 1974. [Google Scholar]
- Lie, S. Theorie der Transformationsgrupprn: Vol I; Chelsea: New York, NY, USA, 1970. [Google Scholar]
- Lie, S. Theorie der Transformationsgrupprn: Vol II; Chelsea: New York, NY, USA, 1970. [Google Scholar]
- Lie, S. Theorie der Transformationsgrupprn: Vol III; Chelsea: New York, NY, USA, 1970. [Google Scholar]
- Ibragimov, N.H. CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws; CRS Press LLC: Boca Raton, FL, USA, 2000. [Google Scholar]
- Bluman, G.W.; Kumei, S. Symmetries of Differential Equations; Springer: New York, NY, USA, 1989. [Google Scholar]
- Stephani, H. Differential Equations: Their Solutions Using Symmetry; Cambridge University Press: New York, NY, USA, 1989. [Google Scholar]
- Olver, P.J. Applications of Lie Groups to Differential Equations; Springer: New York, NY, USA, 1993. [Google Scholar]
- Chesnokov, A.A. Symmetries and exact solutions of the rotating shallow-water equations. Eur. J. Appl. Math. 2009, 20, 461. [Google Scholar] [CrossRef]
- Paliathanasis, A. One-Dimensional Optimal System for 2D Rotating Ideal Gas. Symmetry 2019, 11, 1115. [Google Scholar] [CrossRef]
- Bihlo, A.; Poltavets, N.; Popovych, R.O. Point symmetry group of the barotropic vorticity equation. Chaos 2020, 30, 073132. [Google Scholar] [CrossRef] [PubMed]
- Meleshko, S.V.; Samatova, N.F. Invariant solutions of the two-dimensional shallow water equations with a particular class of bottoms. AIP Conf. Proc. 2019, 2164, 050003. [Google Scholar]
- Ouhadan, A.; Kinami, E.H.E. Lie symmetries analysis of the shallow water equations. Appl. Math. E-Notes 2009, 9, 281. [Google Scholar]
- Paliathanasis, A. Lie Symmetries and Similarity Solutions for Rotating Shallow Water. Z. Naturforschung 2019, 74, 869. [Google Scholar] [CrossRef]
- Dorodnitsyn, V.A.; Kaptsov, E.I. Discrete shallow water equations preserving symmetries and conservation laws. J. Math. Phys. 2021, 62, 083508. [Google Scholar] [CrossRef]
- Dorodnitsyn, V.A.; Kaptsov, E.I. Shallow water equations in Lagrangian coordinates: Symmetries, conservation laws and its preservation in difference models. Commun. Nonlinear Sci. Numer. Simul. 2020, 89, 105343. [Google Scholar] [CrossRef]
- Liu, J.-G.; Zeng, Z.-F.; He, Y.; Ai, G.-P. A class of exact solution of (3+ 1)-dimensional generalized shallow water equation system. Int. J. Nonlinear Sci. Numer. Simul. 2013, 16, 114. [Google Scholar]
- Szatmari, S.; Bihlo, A. Symmetry analysis of a system of modified shallow-water equations. Commun. Nonlinear Sci. Numer. Simul. 2014, 19, 530. [Google Scholar] [CrossRef]
- Exner, F.M. Über die wechselwirkung zwischen wasser und geschiebe in flüssen. Akad. Wiss. Wien Math. Naturwiss. Kl. 1925, 134, 165–204. [Google Scholar]
- Audusse, E.; Chalons, C.; Ung, P. A simple three-wave approximate Riemann solver for the Saint-Venant-Exner equations. Numer. Math. Fluids 2018, 87, 508. [Google Scholar] [CrossRef]
- Siviglia, A.; Vanzo, D.; Toro, E.F. A splitting scheme for the coupled Saint-Venant-Exner model. J. Adv. Water Resour. 2022, 159, 104062. [Google Scholar] [CrossRef]
- Fernández-Nieto, E.D.; Lucas, C.; de Luna, T.M.; Cordier, S. On the influence of the thickness of the sediment moving layer in the definition of the bedload transport formula in Exner systems. Comput. Fluids 2014, 91, 87–106. [Google Scholar] [CrossRef]
- Lyn, D.A.; Altinakar, M. St. Venant–Exner equations for near-critical and transcritical flows. J. Hydraul. Eng. 2002, 128, 579. [Google Scholar] [CrossRef]
- Siviglia, A.; Nobile, G.; Colombini, M. Quasi-Conservative Formulation of the One-Dimensional Saint-Venant-Exner Model. J. Hydraul. Eng. 2008, 134, 1521. [Google Scholar] [CrossRef]
- Hudson, J.; Sweby, P.K. Formulations for Numerically Approximating Hyperbolic Systems Governing Sediment Transport. J. Sci. Comput. 2003, 19, 225. [Google Scholar] [CrossRef]
- Grass, A.J. Sediment Transport by Waves and Currents; University College, Department of Civil Engineering: London, UK, 1981. [Google Scholar]
- Siviglia, A.; Stecca, G.; Vanzo, D.; Zolezzi, G.; Toro, E.F.; Tubino, M. Numerical modelling of two-dimensional morphodynamics with applications to river bars and bifurcations. Adv. Water Resour. 2013, 52, 243. [Google Scholar] [CrossRef]
- Diaz, M.J.C.; Fernandez-Nieto, E.D.; Ferreiro, A.M.; Pares, C. Two-Dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes. Comput. Methods Appl. Mech. Eng. 2009, 198, 2520. [Google Scholar] [CrossRef]
- Berthon, C.; Cordier, S.; Delestre, O.; Le, M.H. An analytical solution of the shallow water system coupled to the Exner equation. C. R. Math. 2012, 350, 183. [Google Scholar] [CrossRef]
- Patera, J.; Sharp, R.T.; Winternitz, P.; Zassenhaus, H. Invariants of real low dimension Lie algebras. J. Math. Phys. 1976, 17, 986. [Google Scholar] [CrossRef]
- Cherniha, R.; Kovalenko, S. Lie symmetries of nonlinear boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 71. [Google Scholar] [CrossRef]
- Ovsiannikov, L.V. Group Analysis of Differential Equations; Academic Press: New York, NY, USA, 1982. [Google Scholar]
0 | 0 | 0 | |||
0 | 0 | 0 | 0 | ||
0 | 0 | 0 | |||
0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Paliathanasis, A. Lie Symmetry Analysis of the One-Dimensional Saint-Venant-Exner Model. Symmetry 2022, 14, 1679. https://doi.org/10.3390/sym14081679
Paliathanasis A. Lie Symmetry Analysis of the One-Dimensional Saint-Venant-Exner Model. Symmetry. 2022; 14(8):1679. https://doi.org/10.3390/sym14081679
Chicago/Turabian StylePaliathanasis, Andronikos. 2022. "Lie Symmetry Analysis of the One-Dimensional Saint-Venant-Exner Model" Symmetry 14, no. 8: 1679. https://doi.org/10.3390/sym14081679
APA StylePaliathanasis, A. (2022). Lie Symmetry Analysis of the One-Dimensional Saint-Venant-Exner Model. Symmetry, 14(8), 1679. https://doi.org/10.3390/sym14081679