An improved third‐order finite difference weighted essentially nonoscillatory scheme for hyperbolic conservation laws. (4th June 2020)

• Record Type: Journal Article
• Authors: Li, Guodong; Li, Xiaogang; Li, Pengfen; Cai, Dandan
• Is Part Of: International journal for numerical methods in fluids. Volume 92:Number 12(2020)
• Page Start: 1753
• Page End: 1777
• Publication Date: 2020-06-04
• Subjects: convergence accuracy -- hyperbolic conservation laws -- smoothness indicators -- WENO scheme; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4847
• Languages: English
• ISSNs: 0271-2091

Abstract

Summary: In this article, we present an improved third‐order finite difference weighted essentially nonoscillatory (WENO) scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is an extension of WENO‐ZQ scheme. However, the global smoothness indicator has a little different from WENO‐ZQ scheme. In this follow‐up article, a convex combination of a second‐degree polynomial with two linear polynomials in a traditional WENO fashion is used to compute the numerical flux at cell boundary. Although the same three‐point information is adopted by the improved third‐order WENO scheme, the truncation errors are smaller than some other third‐order WENO schemes in L ∞ and L 2 norms. Especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. At last, the behavior of improved scheme is proved on a variety of one‐ and two‐dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with other third‐order WENO schemes. Abstract : We presented an improved third‐order finite difference weighted essentially nonoscillatory scheme for achieving the optimal‐order near critical points. Although the same three‐point information is adopted by the improved third‐order WENO scheme, the truncation errors are smaller than other WENO schemes in L ∞ and L 2 norms, especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. The proposed scheme is verified to achieve third‐order accuracy by several benchmarks and the behavior of the present scheme is proved on a variety of one‐ and two‐dimensional standard numerical examples.