Improving the third-order WENO schemes by using exponential polynomial space with a locally optimized shape parameter

Abstract

In this study, we introduce a novel weighted essentially non-oscillatory (WENO) conservative finite-difference scheme that improves the performance of the known third-order WENO methods. To approximate sharp gradients and high oscillations more accurately, we incorporate an interpolation method using a set of exponential (or trigonometric) polynomials with an internal shape parameter. In particular, we propose a method to select a locally optimized parameter such that it leads to an enhanced order of accuracy (i.e., fourth-order) regardless of the issue of critical points. Moreover, we present a new type of (local and global) smoothness measures with exponential vanishing moments, resulting in higher decay rates than traditional indicators. The formula for the proposed nonlinear weights ωk includes an important parameter ε that is employed to avoid nullifying the denominator of the unnormalized weights. This has a crucial effect on the order of accuracy of the WENO scheme, especially in the vicinity of the critical points. In this study, we derive a range of ε that guarantees the improved order of accuracy (i.e., fourth-order). Finally, the numerical results are presented to demonstrate the shock-capturing abilities of the proposed WENO scheme. © 2023 Elsevier Ltd