The shallow water equations and their application to realistic cases

Abstract

The numerical modelling of 2D shallow flows in complex geometries involving transient flow and movable boundaries has been a challenge for researchers in recent years. There is a wide range of physical situations of environmental interest, such as flow in open channels and rivers, tsunami and flood modelling, that can be mathematically represented by first-order non-linear systems of partial differential equations, whose derivation involves an assumption of the shallow water type. Shallow water models may include more sophisticated terms when applied to cases of not pure water floods, such as mud/debris floods, produced by landslides. Mud/debris floods are unsteady flow phenomena in which the flow changes rapidly, and the properties of the moving fluid mixture include stop and go mechanisms. The present work reports on a numerical model able to solve the 2D shallow water equations even including bed load transport over erodible bed in realistic situations involving transient flow and movable flow boundaries. The novelty is that it offers accurate and stable results in realistic problems since an appropriate discretization of the governing equations is performed. Furthermore, the present work is focused on the importance of the computational cost. Usually, the main drawback is the high computational effort required for obtaining accurate numerical solutions due to the high number of cells involved in realistic cases. However, the proposed model is able to reduce computer times by orders of magnitude making 2D applications competitive and practical for operational flood prediction. Moreover our results show that high performance code development can take advantage of general purpose and inexpensive Graphical Processing Units, allowing to run almost 100 times faster than old generation codes in some cases.