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We develop a two-dimensional high-order numerical scheme that exactlypreserves and captures the moving steady states of the shallow water equationswith topography or Manning friction. The high-order accuracy relies on asuitable polynomial reconstruction, while the well-balancedness property isbased on the first-order scheme from Michel-Dansac et. al., 2016 andMichel-Dansac et. al., 2017, extended to two space dimensions. To get bothproperties, we use a convex combination between the high-order scheme and thefirst-order well-balanced scheme. By adequately choosing the convex combinationparameter following a very simple steady state detector, we ensure that theresulting scheme is both high-order accurate and well-balanced. The method isthen supplemented with a MOOD procedure to eliminate the spurious oscillationscoming from the high-order polynomial reconstruction and to guarantee thephysical admissibility of the solution. Numerical experiments show that thescheme indeed possesses the claimed properties. The simulation of the 2011Japan tsunami, on real data, further confirms the relevance of this technique.

A two-dimensional high-order well-balanced scheme for the shallow water equations with topography and Manning friction