Improved third-order WENO scheme with a new reference smoothness indicator

In this article, a new reference smoothness indicator for the third-order WENO scheme is proposed. The main construction idea is to analyze the first derivative of the reconstruction polynomial of the candidate sub-stencil and the first derivative of the reconstruction polynomial of the global stencil. Then calculate the normalized square sum of the $L^2$-norm approximation of the linear convex combination of the first derivative of the reconstruction polynomial of all candidate sub-stencils and the first derivative of the reconstruction polynomial of the global stencil. The new reference smoothness indicator obtained based on the above strategy is denoted as ${\tau }_{Re}$. The newly developed solution is called the WENO-Re solution. A series of one-dimensional and two-dimensional numerical examples show that the new scheme has higher resolution and smaller dissipation compared to several recently improved third-order WENO schemes.