Review on weighting functions of high-order nonlinear weighted methods

After a brief introduction to the historical development of nonlinear schemes for capturing discontinuities, this paper delves into the research on selecting candidate stencils, calculating smoothness indicators and their relationships with nonlinear weighting functions. This review is centered on the fifth order WENO scheme, serving as a representative example of nonlinear weighted schemes. It highlights the crucial requirements for the leading term order of nonlinear weights to preserve the accuracy of these schemes, especially when dealing with candidate stencils of varying widths. This paper underscores the significance of smoothness indicator calculations in ensuring both the accuracy and efficiency of the scheme. Finally, it offers insights and suggestions for future research aimed at further enhancing nonlinear weighted schemes that utilize candidate stencils with non-uniform widths.