Summary
Free-surface gravity flows are stationary points of a functional J when the problem is formulated variationally.
Here we are concerned with the problem of determining the nature of the stationary point, that is, whether it is a minimum, a maximum, a saddle point or whether a singularity occurs. This is a problem of both theoretical and computational importance.
Within a variational approximation of shallow-water type developed by the authors, we prove some new results on the problem. The analysis is carried out by studying the second variation of the functional J and the corresponding Jacobi's equation.
Reference is also made to numerical experiments which confirm the findings. The experiments also suggest that such findings may well extend to flows outside the class of shallow-water flows governed by the model used in the analysis.
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Toro, E.F., O'Carroll, M.J. Types of stationary points in a variational formulation of shallow-water flows. J Eng Math 18, 195–205 (1984). https://doi.org/10.1007/BF00039188
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DOI: https://doi.org/10.1007/BF00039188