A New Alternative WENO Scheme Based on Exponential Polynomial Interpolation with an Improved Order of Accuracy

Abstract

In this study, we present a new alternative formulation of a conservative weighted essentially non-oscillatory (WENO) scheme that improves the performance of the known fifth-order alternative WENO (AWENO) schemes. In the formulation of the fifth-order AWENO scheme, the numerical flux can be written in two terms: a low-order flux and a high-order correction flux. The low-order numerical flux is constructed by a fifth-order WENO interpolator, and the high-order correction flux includes terms of the second and fourth derivatives, yielding the sixth-order truncation error. Noticing the difference in the convergence rates between these two approximations, this study first aims to fill the accuracy gap by enhancing the approximation order of the low-order numerical flux. To this end, the WENO interpolator for the low-order term is implemented using exponential polynomials with a shape parameter. Selecting a locally optimized shape parameter, the proposed WENO interpolator achieves an additional order of improvement, resulting in the overall sixth order of accuracy of the final reconstruction, under the same fifth-order AWENO framework. In addition, since a linear approximation to the high-order correction term may cause some oscillations in the vicinity of strong shocks, we present a new strategy for the limiting procedure to deal with the second derivative term in the high-order correction flux. Several numerical results for the well-known benchmark test problems confirm the reliability of our AWENO method.