Optimal Definition of the Nonlinear Weights in Multidimensional Central WENOZ Reconstructions

Central $\mathsf{WENO}$ reconstruction procedures have shown very good performance in finite volume and finite difference schemes for hyperbolic conservation and balance laws in one and higher space dimensions on different types of meshes. Their most recent formulations include $\mathsf{WENOZ}$-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 multidimensional Jiang--Shu smoothness indicators that are useful for the rigorous design of $\mathsf{CWENOZ}$ schemes, which are in addition to those considered in this paper. Next, we introduce the optimal definition of $\tau$ for the one-dimensional $\mathsf{CWENOZ}$ schemes and for one example of two-dimensional $\mathsf{CWENOZ}$ reconstruction. Numerical experiments of one- and two-dimensional test problems show the correctness of the analysis and the good performance of the new schemes.