Improved third-order WENO scheme with new reference smoothness indicator

Abstract

In this article, a new reference smoothness indicator for third-order WENO schemes is proposed.The main construction idea is: by analyzing the relationship between the derivatives of the reconstruction polynomials of the candidate stencils and the reconstruction polynomials of the global stencil,and then calculating the $L^2$-norm approximation of the linear combination of the derivatives of the reconstruction polynomials of all candidate stencils and the derivatives of the reconstruction polynomials of the global stencil. Furthermore,a new reference smoothness indicator (τ_{Re})is obtained,the newly developed scheme is called WENO-Re scheme. A series of one-dimensional and two-dimensional numerical examples show that the new scheme has higher resolution and less dissipation than several recently improved third-order WENO schemes.