We discussed in Chapter 12 that de Saint Venant equations are nonlinear partial differential equations for which a closed form solution is not available except for very simplified cases. In Chapter 13, we briefly presented several numerical methods that may be used for their integration. Of these methods, the finite-difference methods have been utilized very extensively; details of some of these methods are outlined in this chapter. Either a conservation or nonconservation form of the governing equations may be used in some methods whereas only one of these forms may be used in others. A conservation form should be preferred, since it conserves various quantities better and it simulates the celerity of wave propagation more accurately than the nonconservation form [Cunge et al., 1980; Miller and Chaudhry, 1989].
We first discuss a number of commonly used terms. Then, a number of explicit and implicit finite-difference methods are presented and the inclusion of boundary conditions in these methods is outlined. The consistency of a numerical scheme is briefly discussed and the stability conditions are then derived. The results computed by different schemes are compared.
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(2008). Finite-Difference Methods. In: Open-Channel Flow. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-68648-6_14
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