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

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 ω_k 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.