Semi-analytical solutions of shallow water waves with idealised bottom topographies

References

• Adomian, G., A review of the decomposition method and some recent results for nonlinear equations. Math. Comput. Modell. 1990, 13, 17–43.
   Web of Science ®Google Scholar
• Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Vol. 60, 2013 (Dordrecht: Springer Science & Business Media).
   Google Scholar
• Ball, F.K., Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. J. Fluid Mech. 1963, 17, 240–256.
   Web of Science ®Google Scholar
• Ball, F.K., An exact theory of simple finite shallow water oscillations on a rotating earth. In Hydraulics and Fluid Mechanics, pp. 293–305, 1964 (Oxford: Pergamon Press LTD).
   Google Scholar
• Ball, F.K., The effect of rotation on the simpler modes of motion of a liquid in an elliptic paraboloid. J. Fluid Mech. 1965, 22, 529–545.
   Web of Science ®Google Scholar
• Bihlo, A., Poltavets, N. and Popovych, R.O., Lie symmetries of two-dimensional shallow water equations with variable bottom topography. Chaos 2020, 30, 073132.
   Google Scholar
• Bila, N., Mansfield, E.L. and Clarkson, P.A., Symmetry group analysis of the shallow water and semi-geostrophic equations. Q. J. Mech. Appl. Math. 2006, 59, 95–123.
   Web of Science ®Google Scholar
• Bollermann, A., Noelle, S. and Lukáčová-Medvid'ová, M., Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 2011, 10, 371–404.
   Web of Science ®Google Scholar
• Bristeau, M.O., Di Martino, B., Mangeney, A., Sainte-Marie, J. and Souillé, F., Some analytical solutions for validation of free surface flow computational codes. J. Fluid Mech. 2021, 913, A17.
   Web of Science ®Google Scholar
• Chesnokov, A.A., Symmetries and exact solutions of the rotating shallow-water equations. Eur. J. Appl. Math. 2009, 20, 461–477.
   Web of Science ®Google Scholar
• Chesnokov, A.A., Properties and exact solutions of the equations of motion of shallow water in a spinning paraboloid. J. Appl. Math. Mech. 2011, 75, 350–356.
   Google Scholar
• Clark, A.D. and Herron, I.H., Improved bounds on linear instability of barotropic zonal flow within the shallow water equations. Geophys. Astrophys. Fluid Dyn. 2013, 107, 328–352.
   Web of Science ®Google Scholar
• Clarkson, P.A. and Kruskal, M.D., New similarity reductions of the Boussinesq equation. J. Math. Phys. 1989, 30, 2201–2213.
   Web of Science ®Google Scholar
• Currò, C., Some new exact solutions to the nonlinear shallow-water wave equations via group analysis. Meccanica 1989, 24, 26–35.
   Google Scholar
• Cushman-Roisin, B., Exact analytical solutions for elliptical vortices of the shallow-water equations. Tellus A 1987, 39, 235–244.
   Google Scholar
• Cushman-Roisin, B., Heil, W.H. and Nof, D., Oscillations and rotations of elliptical warm-core rings. J. Geophys. Res.: Oceans 1985, 90, 11756–11764.
   Web of Science ®Google Scholar
• Delestre, O., Lucas, C., Ksinant, P.A., Darboux, F., Laguerre, C., Vo, T.N.T., James, F. and Cordier, S., SWASHES: a compilation of shallow water analytic solutions for hydraulic and environmental studies. Int. J. Numer. Methods Fluids 2013, 72, 269–300.
   Web of Science ®Google Scholar
• Ern, A., Piperno, S. and Djadel, K., A well-balanced Runge–Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. Int. J. Numer. Methods Fluids 2008, 58, 1–25.
   Web of Science ®Google Scholar
• Gallardo, J.M., Parés, C. and Castro, M., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 2007, 227, 574–601.
   Web of Science ®Google Scholar
• Iacono, R., Analytic solutions to the shallow water equations. Phys. Rev. E 2005, 72, 017302.
   Google Scholar
• Kafiabad, H.A., Vanneste, J. and Young, W.R., Interaction of near-inertial waves with an anticyclonic vortex. J. Phys. Oceanogr. 2021, 51, 2035–2048.
   Web of Science ®Google Scholar
• Kesserwani, G. and Liang, Q., Locally limited and fully conserved RKDG2 shallow water solutions with wetting and drying. J. Sci. Comput. 2012, 50, 120–144.
   Web of Science ®Google Scholar
• Levi, D., Nucci, M., Rogers, C. and Winternitz, P., Group theoretical analysis of a rotating shallow liquid in a rigid container. J. Phys. A: Math. Gen. 1989, 22, 4743.
   Google Scholar
• Li, M., Guyenne, P., Li, F. and Xu, L., A positivity-preserving well-balanced central discontinuous Galerkin method for the nonlinear shallow water equations. J. Sci. Comput. 2017, 71, 994–1034.
   Web of Science ®Google Scholar
• Matskevich, N.A. and Chubarov, L.B., Exact solutions to shallow water equations for a water oscillation problem in an idealized basin and their use in verifying some numerical algorithms. Numer. Anal. Appl. 2019, 12, 234–250.
   Web of Science ®Google Scholar
• McKiver, W.J., Balanced ellipsoidal vortex at finite Rossby number. Geophys. Astrophys. Fluid Dyn. 2020, 114, 453–480.
   Web of Science ®Google Scholar
• Meleshko, S.V., Complete group classification of the two-dimensional shallow water equations with constant Coriolis parameter in Lagrangian coordinates. Commun. Nonlinear Sci. Numer. Simul. 2020, 89, 105293.
   Google Scholar
• Meleshko, S.V. and Samatova, N.F., Group classification of the two-dimensional shallow water equations with the beta-plane approximation of Coriolis parameter in Lagrangian coordinates. Commun. Nonlinear Sci. Numer. Simul. 2020, 90, 105337.
   Google Scholar
• Nikolos, I.K. and Delis, A.I., An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography. Comput. Methods Appl. Mech. Eng. 2009, 198, 3723–3750.
   Web of Science ®Google Scholar
• Pedlosky, J., Geophysical Fluid Dynamics, 2013 (New York: Springer Science & Business Media).
   Google Scholar
• Rogers, C., Generation of invariance theorems for nonlinear boundary-value problems in shallow water theory: an application of MACSYMA. In Numerical and Applied Mathematics, IMACS Meeting Proceedings, Paris, 1989, pp. 69–74 (Paris: IMACS Meeting Proceedings).
   Google Scholar
• Rogers, C., Elliptic warm-core theory: the pulsrodon. Phys. Lett. A 1989, 138, 267–273.
   Web of Science ®Google Scholar
• Sachdev, P.L., Palaniappan, D. and Sarathy, R., Regular and chaotic flows in paraboloidal basins and eddies. Chaos, Solitons Fractals 1996, 7, 383–408.
   Web of Science ®Google Scholar
• Sampson, J., Easton, A. and Singh, M., Moving boundary shallow water flow above parabolic bottom topography. ANZIAM J. 2005, 47, 373–387.
   Google Scholar
• Shapiro, A., Nonlinear shallow-water oscillations in a parabolic channel: exact solutions and trajectory analyses. J. Fluid Mech. 1996, 318, 49–76.
   Web of Science ®Google Scholar
• Sun, C., High-order exact solutions for pseudo-plane ideal flows. Phys. Fluids 2016, 28, 083602.
   Google Scholar
• Thacker, W.C., Irregular grid finite-difference techniques: simulations of oscillations in shallow circular basins. J. Phys. Oceanogr. 1977, 7, 284–292.
   Web of Science ®Google Scholar
• Thacker, W.C., Some exact solutions to the nonlinear shallow-water wave equations. J. Fluid Mech. 1981, 107, 499–508.
   Web of Science ®Google Scholar
• Tsang, Y.K. and Dritschel, D.G., Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation. J. Fluid Mech. 2015, 762, 196–231.
   Web of Science ®Google Scholar
• Vallis, G.K., Atmospheric and Oceanic Fluid Dynamics, 2017 (Cambridge: Cambridge University Press).
   Google Scholar
• Vallis, G.K., Essentials of Atmospheric and Oceanic Dynamics, 2019 (Cambridge: Cambridge University Press).
   Google Scholar
• Wintermeyer, N., Winters, A.R., Gassner, G.J. and Warburton, T., An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. J. Comput. Phys. 2018, 375, 447–480.
   Web of Science ®Google Scholar