A modified fifth-order WENO-Z scheme based on the weights of the reformulated adaptive order WENO scheme

A modified fifth-order WENO-Z scheme is proposed by analogy with the non-normalized weights of the reformulated fifth-order adaptive order (AO) WENO scheme. We show that if the original fifth-order WENO-AO scheme is rewritten as the form of the conventional WENO combination, the resulting non-normalized weights can be divided into three parts: a constant one term, a local stencil smoothness measure term and a global stencil smoothness measure term. In order to make use of the latter two terms for constructing a modified WENO-Z scheme with enhanced performance, we change the form of the third term and introduce an adaptive scaling factor to adjust the contributions from the second and third terms. Numerical examples show that the modified fifth-order WENO-Z scheme has the advantage of high resolution in smooth regions and sharp capturing of discontinuities, and it can obtain evidently better results for shocked flows with small-scale structures compared with the recently developed WENO-Z+ and WENO-Z+M schemes.