Perturbed Polynomial With Multiple Free‐Parameters Reconstructed WENO Schemes

Abstract

The classical WENO schemes perform well for most flow field simulations, they may encounter the ‘Cannikin Law’ trap, that is, the lowest accuracy order of the scheme may have a significant influence on the simulation. In this article, a novel WENO scheme (termed HPWENO) with improved convergence order is proposed to alleviate this issue. The research in this article is structured around three key steps: Firstly, the stencil is classified as either smooth stencil or non‐smooth stencil by using the classification strategy of the hybrid WENO scheme. Secondly, perturbed polynomial reconstruction with double free‐parameters is proposed. Finally, the new reconstruction coefficients containing multiple free‐parameters, built on the classical fifth‐order WENO schemes, are obtained by using the perturbed polynomial reconstruction. Compared to the fifth‐order WENO schemes, a maximum two‐order of accuracy improvement in candidate stencils and one‐order of accuracy improvement in global stencil can be achieved by adaptively adjusting the values of these free‐parameters, resulting in sixth‐order accuracy in global stencil and fifth‐order accuracy in candidate stencils. Compared to the classical fifth‐order WENO5‐Z scheme and the WENO‐AO(5,3) scheme, numerical examples show that the HPWENO schemes have higher convergence ratio, provide sharper solution profiles near discontinuities, and perform well in resolving small‐scale structures. Compared to the sixth‐order WENO‐CU6 scheme and the seventh‐order WENO7‐Z scheme, the proposed HPWENO schemes outperform the two schemes in resolving the small‐scale vortex of two‐dimensional issues, and it saves approximately 15% and 25% of computational resources, respectively.