In this paper a problem of multiple solutions of steady gradually varied flow equation in the form of the ordinary differential energy equation is discussed from the viewpoint of its numerical solution. Using the Lipschitz theorem dealing with the uniqueness of solution of an initial value problem for the ordinary differential equation it was shown that the steady gradually varied flow equation can have more than one solution. This fact implies that the nonlinear algebraic equation approximating the ordinary differential energy equation, which additionally coincides with the wellknown standard step method usually applied for computing of the flow profile, can have variable number of roots. Consequently, more than one alternative solution corresponding to the same initial condition can be provided. Using this property it is possible to compute the water flow profile passing through the critical stage.
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Abbot M.B. 1979. Computational Hydraulics: Elements of the Theory of Free-Surface Flows. Pitman London.
Artichowicz W. 2012. Numerical modeling of steady gradually varied flow in open channels. PhD Thesis. Gdansk University of Technology Faculty of Civil and Environmental Engineering Gdansk.
Artichowicz W. Szymkiewicz R. 2013. Properties of one dimensional open-channel steady flow equations. In: Proc. 13th International Symposium on Water Management and Hydraulic Engineering Bratislava Slovakia.
Ascher U.M. Petzold L.R. 1998. Computer Methods for Ordinary Differential Equations and Difference-Algebraic Equations. SIAM Philadelphia.
Castro-Orgaz O. Giráldez J.V. Ayuso J.L. 2008. Energy and momentum under critical flow conditions. Journal of Hydraulic Research 46 6 844-848.
Chanson H. 2004. The Hydraulics of Open Channel Flow: An Introduction. Second Edition. Elsevier Oxford.
Chippada S. Ramaswamy B. Wheeler M.F. 1994. Numerical simulation of the hydraulic jump. International Journal for Numerical Methods in Engineering 37 8 1381-1397.