A class of high-order improved fast weighted essentially non-oscillatory schemes for achieving optimal order at any critical points

Abstract

The weighted essentially non-oscillatory (WENO) scheme is one of the most popular shock-capturing schemes, and constructing a more efficient and higher-order WENO scheme has always been an intention of optimization design. In the general WENO reconstruction framework, the smoothness indicator plays an important role in identifying whether the sub-stencils are in discontinuous or smooth regions. However, the classical smoothness indicator is the most expensive one in the whole reconstruction algorithm, and its computational complexity increases sharply with the improvement of the accuracy order. Therefore, a class of efficient and superior WENO schemes called improved fast WENO (IFWENO) are proposed based on the fast WENO (FWENO). To improve efficiency, the smoothness indicator of the IFWENO scheme is simplified from the traditional version, and the nonlinear weight calculation method is modified. The parameter ε is carefully designed to obtain the superior property that the accuracy of the spatial derivatives will not degrade at any order critical point in smooth regions. The reason for the instability occurring in the high-order FWENO is revealed, and the parameter p is likewise specifically selected to improve robustness at discontinuities. The excellent multi-scale resolution of the proposed IFWENO scheme is proven by theoretical analyses and numerical experiments. Through several typical examples, the consistently high accuracy and efficiency of the designed scheme in both smooth and discontinuous regions are verified.