High order weighted essentially non-oscillatory WENO-ZN schemes for hyperbolic conservation laws

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In Shen et al. (2020), the authors have proposed a novel weighting method to construct the fifth-order WENO-ZN scheme to improve the accuracy at the second-order critical point. Its basic idea is that, the square of the fourth-order undivided difference on the global five-point stencil used by the fifth-order WENO scheme is suggested as the global smoothness indicator. To keep the ENO property and enhance robustness for resolving shock waves, the constant 1 used to calculate the un-normalized weights in the original WENO-Z schemes is replaced by an adaptive function, which can approach a small value if the global stencil contains a discontinuity or approach a large value if the solution is smooth enough. The fifth-order WENO-ZN scheme can obtain fifth order accuracy at both the first- and second-order critical points. However, limited by the smoothness indicators, the scheme cannot improve the convergence rate at the third-order and above critical points. In this paper, we extend the idea of the fifth-order WENO-ZN scheme to construct higher-order WENO-ZN schemes and investigate their performance. Numerical experiments show that the (2r−1)th-order (r≥3) WENO-ZN schemes are robust for capturing shock waves and can improve the accuracy order in smooth regions including the maximum (2r−4)th-order critical points.