  We present a novel arbitrary high order accurate central WENO spatial
reconstruction procedure (CWENO) for the solution of nonlinear systems of
hyperbolic conservation laws on fixed and moving unstructured simplex meshes in
two and three space dimensions. Starting from the given cell averages of a
function on a triangular or tetrahedral control volume and its neighbors, the
nonlinear CWENO reconstruction yields a high order accurate and essentially
non-oscillatory polynomial that is defined everywhere in the cell. Compared to
other WENO schemes on unstructured meshes, the total stencil size is the
minimum possible one, as in classical point-wise WENO schemes of Jiang and Shu.
However, the linear weights can be chosen arbitrarily, which makes the
practical implementation on general unstructured meshes particularly simple. We
make use of the piecewise polynomials generated by the CWENO reconstruction
operator inside the framework of fully discrete and high order accurate
one-step ADER finite volume schemes on fixed Eulerian grids as well as on
moving Arbitrary-Lagrangian-Eulerian (ALE) meshes. The computational efficiency
of the high order finite volume schemes based on the new CWENO reconstruction
is tested on several two- and three-dimensional systems of hyperbolic
conservation laws and is found to be more efficient in terms of memory
consumption and computational efficiency with respect to classical WENO
reconstruction schemes on unstructured meshes. We also provide evidence that
the new algorithm is suitable for implementation on massively parallel
distributed memory supercomputers, showing two numerical examples run with more
than one billion degrees of freedom in space. To our knowledge, at present
these are the largest simulations ever run with unstructured WENO finite volume
schemes.
