Perturbed polynomial reconstructed seventh-order hybrid WENO scheme with improved accuracy and resolution

Non-polynomial reconstructions can be employed to enhance the performance of the WENO-type schemes by optimizing the inherent hyper-parameter. In contrast to the non-polynomial RBF-based and the Gauss-Kriging reconstructions, the perturbed polynomial reconstruction exhibits good portability and expandability. In this work, a novel seventh-order WENO scheme, denoted as the HPWENO7 scheme, is proposed by incorporating the concept of the perturbed polynomial reconstructions into the standard seventh-order WENO7-JS scheme (Jiang and Shu, 1996, [6]). Firstly, a refined troubled cell indicator is developed to categorize the global stencils as either smooth or non-smooth. Subsequently, perturbed polynomial reconstruction with double free-parameters, the values of which can be adjusted automatically according to the features of local regions, is developed to optimize the fluxes within the four-point candidate stencils. Adaptive optimization of the free-parameter values enables a minimum one-order improvement in accuracy. Finally, the novel HPWENO7 scheme is proposed by combining the seventh-order upstream central scheme for smooth stencils with the perturbed polynomial reconstruction optimized candidate fluxes for non-smooth stencils. Numerical examples show that the HPWENO7 scheme achieves fifth-order of accuracy in the four-point candidate stencils, providing sharper solutions for discontinuities and significantly higher resolution for small-scale vortex structures around the discontinuities.