Novel Hybrid Unequal-Sized WENO Scheme Employing Trigonometric Polynomials for Solving Hyperbolic Conservation Laws on Structured Grids

This study presents a novel fifth-order unequal-sized trigonometric weighted essentially non-oscillatory (US-TWENO) scheme and a novel hybrid US-TWENO (HUS-TWENO) scheme with a novel troubled cell indicator in a finite difference framework to address hyperbolic conservation laws on structured grids. Firstly, we propose three unequal-degree reconstruction polynomials in the new trigonometric polynomial space to devise a novel fifth-order US-TWENO scheme. Then, we devise a novel troubled cell indicator capable of accurately identifying troubled cells containing strong discontinuities: the existence of extreme points of the trigonometric polynomials within the smallest interval (the target cell itself) is determined by whether the estimated minimum and maximum values of their derivative trigonometric polynomials have opposite signs. To the best of our knowledge, this is the first troubled cell indicator devised specifically within the target cell interval. The HUS-TWENO scheme is improved, offering greater efficiency, lower dissipation, and higher resolution. Numerical experiments demonstrate the effectiveness of the HUS-TWENO scheme.