A New Hybrid Trigonometric WENO Scheme for Hamilton-Jacobi Equations

In this paper, we introduce a new hybrid weighted essentially non-oscillatory (WENO) scheme grounded in trigonometric polynomials for the solution of Hamilton-Jacobi equations. This innovative approach utilizes trigonometric polynomial reconstruction as an alternative to the conventional algebraic polynomial reconstruction. Notably, the proposed scheme demonstrates reduced truncation errors in smooth regions and enhanced resolution in nonsmooth regions, outperforming the classical trigonometric WENO scheme — cf. [J. Zhu and J.X. Qiu, $Commun. Comput. Phys. 8$ (2010)]. In particular, we have designed a novel streamlined hybrid strategy aimed at improving the computational efficiency of the scheme, which surpasses the classical WENO scheme in this regard. A comprehensive suite of numerical experiments has been undertaken to demonstrate the merits of the proposed scheme.