Improved Three-level Order-Adaptive WENO Scheme

Abstract

Weighted essentially non-oscillatory schemes (WENO) are reconstruction schemes having higher-order and high-resolution properties. This class of schemes can adapt its order based on the smoothness of the solution. The classical fifth-order WENO scheme is a two-level scheme that can achieve fifth-order accuracy for smooth solutions and thirdorder accuracy for non-smooth solutions. However, they cannot achieve fourth-order accuracy. In this work, we propose a threelevel scheme that can achieve third, fourth, and fifth-order accuracies. A new weighing function is introduced for achieving fourth-order accuracy by using the standard substencils of classical WENO schemes. The scheme is tested on variety of 1-D and 2-D problems based on the gas dynamic equations. The proposed scheme exhibit excellent shock resolving property and high computational accuracy. For the shock-shear layer interaction test case, the present scheme found to be five times computationally more economical than the WENO-JS scheme.