Optimal definition of the nonlinear weights in multidimensional Central WENOZ reconstructions

Central WENO reconstruction procedures have shown very good performances in finite volume and finite difference schemes for hyperbolic conservation and balance laws in one and more space dimensions, on different types of meshes. Their most recent formulations include WENOZ-type nonlinear weights, but in this context a thorough analysis of the global smoothness indicator $\tau$ is still lacking. In this work we first prove results on the asymptotic expansion of one- and multi-dimensional Jiang-Shu smoothness indicators that are useful for the rigorous design of a CWENOZ schemes, also beyond those considered in this paper. Next, we introduce the optimal definition of $\tau$ for the one-dimensional CWENOZ schemes and for one example of two-dimensional CWENOZ reconstruction. Numerical experiments of one and two dimensional test problems show the correctness of the analysis and the good performance of the new schemes.