Improved weighted essentially non-oscillatory schemes with modified stencil approximation

In this article, a new modified stencil approximation for weighted essentially non-oscillatory (WENO) schemes is proposed to reduce numerical dissipation of classical weighted essentially non-oscillatory (WENO-JS) schemes. Since the addition of high-order terms pk(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^k(x)$$\end{document} improves the accuracy of approximation polynomials of candidate stencils, the approximate accuracy of numerical fluxes of candidate stencils in classical WENO scheme is improved. In addition, the corresponding candidate fluxes are calculated, which can make the resulting scheme (called WENO-MS) achieve optimal convergence order in smooth regions including first-order critical points. A series of numerical examples are presented to demonstrate the performance of the new scheme. The numerical results show that the proposed WENO-MS schemes provide a comparable or higher resolution of fine smooth structures compared with the WENO-JS and WENO-Z schemes.