A novel high resolution fifth-order weighted essentially non-oscillatory scheme for solving hyperbolic equations

Abstract

Weighted essentially non-oscillator (WENO) scheme is popular in solving hyperbolic conservation equations. In this work, a novel high resolution fifth-order WENO scheme is presented to improve the performance for capturing of discontinuities and complex structures. A local smoothness indicator ISk with compact form is adopted to the WENO scheme of Borges (WENO-Z). Then, a new global smooth indicator η containing information of all sub-templates is constructed to improve the resolution. Furthermore, the nonlinear weight strategy is optimized, and a new term coefficient λ is proposed, which can adjust the weight adaptively with the change of flow field. The accuracy tests verify that the proposed scheme can achieve the optimal order convergence at the critical points of orders 0, 1, and 2, and the weight analysis indicates that the proposed scheme increases the weights of less smooth sub-templates and gives full play to the potential advantages of WENO-Z. Finally, several typical numerical tests show the proposed scheme can significantly improve the resolution and robustness of WENO scheme. This proposed WENO scheme can capture the discontinuities more sharply and can clearly identify vortex structures and fine vesicles compared with other three WENO schemes.