Abstract
Abstract Free surface flows often appear in ocean, engineering and atmospheric modelling. In many applications involving unsteady water flows where the horizontal length scale is much greater than the vertical length scale, the shallow water equations are commonly used to model these flows. Research on effective and accurate numerical methods for their solutions has attracted great attention in the past two decades. In this chapter, we review some work on designing positivity-preserving and well-balanced methods for solving the shallow water equations with a nonflat bottom topography. Some shallow water-related models, including the shallow water flows through channels with irregular geometry, the shallow water equations on the sphere and the two-layer shallow water equations, and their numerical approximation will also be presented.
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