An efficient three-level weighted essentially non-oscillatory scheme for hyperbolic equations

An improved version of the three-level order-adaptive weighted essentially non-oscillatory (WENO-OA) scheme introduced in Neelan et al. (Results Appl Math 12:100217, 2021) is presented. The dependence of the WENO-OA scheme on the smoothness indicators of the Jiang-Shu WENO (WENO-JS) scheme is replaced with a new smoothness estimator with a smaller computational cost. In the present scheme, the smoothness indicator is only used to identify the smooth and non-smooth sub-stencils of the WENO scheme. The direct connection between the final WENO weights and smoothness indicators is decoupled so that it can exactly satisfy the Taylor expansion, which improves the accuracy of the scheme. The novel scheme denoted WENO-OA-I, is a three-level scheme because it can achieve the order of accuracy from three to five, while the classical scheme only achieves either the third or fifth order of accuracy. As a consequence of this property, the present scheme exhibits improved convergence rates. The performance of the new scheme is tested for hyperbolic equations with discontinuous solutions. The present scheme is up to 5.4 times computationally less expensive than the classical schemes.