  In this paper, we present a novel second-order accurate
Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving
nonconforming polygonal grids, in order to avoid the typical mesh distortion
caused by shear flows in Lagrangian-type methods. In our new approach the
nonconforming element interfaces are not defined by the user, but they are
automatically detected by the algorithm if the tangential velocity difference
across an element interface is sufficiently large. The grid nodes that are
sufficiently far away from a shear wave are moved with a standard node solver,
while at the interface we insert a new set of nodes that can slide in a
nonconforming manner. In this way, the elements on both sides of the shear wave
can move with a different velocity, without producing highly distorted
elements. The core of the proposed method is the use of a space-time
conservation formulation in the construction of the final finite volume scheme,
which completely avoids the need of an additional remapping stage and ensures
that the geometric conservation law (GCL) is automatically satisfied. Moreover,
the mesh quality remains high and, as a direct consequence, also the time step
remains almost constant in time, even for highly sheared vortex flows. The
accuracy of the new scheme has been further improved by incorporating a special
well balancing technique that is able to maintain particular stationary
solutions of the governing PDE system up to machine precision. A large set of
numerical tests has been carried out in order to check the accuracy and the
robustness of the new method. In particular we have compared the results for a
steady vortex in equilibrium solved with a standard conforming ALE method and
with our new nonconforming ALE scheme, to show that the new nonconforming
scheme is able to avoid mesh distortion even after very long simulation times.
