A perturbational weighted essentially non-oscillatory scheme. (30th August 2018)

• Record Type: Journal Article
• Authors: Zeng, Fangjun; Shen, Yiqing; Liu, Shengping
• Is Part Of: Computers & fluids. Volume 172(2018)
• Page Start: 196
• Page End: 208
• Publication Date: 2018-08-30
• Subjects: WENO scheme -- Numerical perturbation method -- Necessary and sufficient conditions -- Low-dissipation; Fluid dynamics -- Data processing -- Periodicals; 532.050285
• DOI: 10.1016/j.compfluid.2018.07.003
• Languages: English
• ISSNs: 0045-7930

Abstract

Highlights: A corollary about the accuracy of a kind of conservative schemes is generalized. A new WENO scheme is constructed by using the perturbed candidate fluxes. The new scheme relaxes the necessary and sufficient conditions for the fifth-order convergence. The new scheme is fifth-order accurate at critical points. The new scheme provides a novel method to decrease the numerical dissipation of traditional WENO schemes. Abstract: In this paper, based on the perturbed fluxes of all candidate fluxes used in the traditional fifth-order WENO scheme, a fifth-order accurate perturbational weighted essentially non-oscillatory (P-WENO) scheme is developed. First, a corollary about the accuracy of a kind of conservative schemes is generalized and proved. Then, based on the corollary and the idea of numerical perturbation, the perturbed fluxes, which are one order higher than the traditional candidate ones of the fifth-order WENO scheme, are obtained. Furthermore, we derive the necessary and sufficient conditions for the fifth-order convergence of the new weighted scheme constructed by using the new perturbed fluxes and find that they are one order lower than those derived by Henrick et al. for the traditional fifth-order WENO scheme. Thus, the new weighted scheme, which uses the same weights of the WENO-Z scheme and the perturbed fluxes, can meet the necessary and sufficient condition for fifth-order convergence even at critical points. The resulted P-WENO scheme actually provides a novel method to decrease the numerical dissipation of traditional WENO schemes. Numerical examples are presented to verify the accuracy, robustness and low-dissipation of the new scheme.