An improved WENO-ZA scheme achieving the optimal convergence order at critical points

To further enhance the numerical accuracy and convergence accuracy of the WENO-ZA scheme, a novel high-order global smoothness indicator is designed by introducing parameters to the classical local smoothness indicator. This enable the scheme to achieve fifth-order convergence accuracy at both the first-order and second-order critical points in the smooth region. Additionally, the local smoothness indicator is utilized to improve the adaptive function. Compared with the original adaptive function, it tends to a larger value at discontinuities to maintain the basic non-oscillatory property of the scheme, while switching to a smaller value in smooth regions to reduce the numerical dissipation of the scheme. Numerical experiments demonstrate that the improved scheme has significant improvements in convergence accuracy at critical points, numerical accuracy, and shock capturing.