A family of efficient high-order hybrid finite difference schemes based on WENO schemes

摘要

This work aims at constructing efficient high-order hybrid schemes to resolve both smooth and discontinuous regions of flow fields. To this end, a new kind of smoothness indicator, referred to as global smoothness indicators in this article, is designed first. The new smoothness indicators contain only simple difference operators and, therefore, are computationally very inexpensive. A family of high-order hybrid schemes is constructed based on the new smoothness indicators. In this kind of hybrid scheme, classic weighted essentially non-oscillatory (WENO) schemes with work as a sub-scheme for capturing the discontinuities, while the corresponding linear optimal schemes of the WENO work as another sub-scheme for resolving the smooth region. It is shown theoretically that the convergence rates of the hybrid schemes always can reach the optimal order as the mesh is refined past a limit resolution. The robustness, high efficiency and high resolution of the hybrid schemes are demonstrated through several representative numerical examples.