Multi-level WENO schemes with an adaptive characteristic-wise reconstruction for system of Euler equations

Abstract

The Weighted Essentially Non-Oscillatory (WENO) scheme is an accurate and robust reconstruction procedure to simulate compressible flows, especially in the presence of discontinuities in the solution. The recently introduced multi-level WENO schemes like WENO-ZQ (Zhu & Qiu, 2016), WENO-AO (Balsara et al. 2016), WENO-ZS (Zhu & Shu 2019) provide better resolution of the solution near discontinuities and accuracy for smooth problems than classical WENO scheme. Despite having better accuracy, a multi-level WENO scheme is not computationally efficient as it involves the computation of an extra smoothness indicator as compared to the classical WENO scheme. In this article, we have proposed two new algorithms of multi-level schemes for system of Euler equations, which retains the accuracy and resolution of multi-level schemes while being less costly. The first algorithm involves identifying the troubled-cells and employing the characteristic-wise WENO reconstruction in the troubled-cells to avoid spurious oscillations and component-wise WENO reconstruction in smooth regions. While in the second algorithm, the latter part is replaced by the component-wise finite difference upwind reconstruction. The important aspect of the new algorithm is the troubled-cell indicator, which has been developed utilizing the information of the extra smoothness indicator of multi-level schemes. The present algorithms are independent of order of multi-level WENO schemes. To demonstrate the new idea, we have considered fifth order WENO-AO(5,3), WENO-ZQ, and WENO-ZS schemes. Several 1D & 2D numerical experiments are performed to demonstrate the efficiency of proposed algorithms and its performance is compared with multi-level schemes. The proposed algorithms are efficient and take 15%–75% less computational time than multi-level schemes while retaining the advantageous features of multi-level schemes.