  High order finite volume schemes for conservation laws are very useful in
applications, due to their ability to compute accurate solutions on quite
coarse meshes and with very few restrictions on the kind of cells employed in
the discretization. For balance laws, the ability to approximate up to machine
precision relevant steady states allows the scheme to compute accurately, also
on coarse meshes, small perturbations of such states.
  In this paper we propose third and fourth order accurate finite volume
schemes for the shallow water equations. The schemes have the well-balanced
property thanks to a path-conservative approach applied to an appropriate
non-conservative reformulation of the equations. High order accuracy is
achieved by designing truly two-dimensional reconstruction procedures of the
CWENO type.
  The novel schemes are tested for accuracy, well-balancing and shown to
maintain posivity of the water height on wet/dry transitions. Finally they are
applied to simulate the Tohoku 2011 tsunami event.
