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