A new family of downwind-limited, scale-invariant WENO schemes with optimal accuracy

Abstract

Despite improvement over decades, there are still at least two key issues not fully solved in Weighted Essentially Non-oscillatory (WENO) schemes. Firstly, most WENO schemes hardly satisfy the scale-invariant property, which undermines their robustness and accuracy when simulating flows involving different scales such as reactive multi-component flows and plasma. Secondly, most of existing Essentially Non-oscillatory (ENO) constraints are not sufficient and necessary conditions to achieve optimal accuracy. Therefore, in this work, we design a new family of WENO schemes based on a three-cell stencil to address these two issues. The new WENO non-linear weights are designed as bounded asymptotic functions which are dependent on cell Normalised Volume Integrated Average (NVIA) values and a shape parameter θ. Thus, the new WENO weighting functions are scale-independent. The shape parameter θ controls the errors between the nonlinear weight and the ideal weight. As the value of θ increases, the new weighting functions become more accurate but also become more likely to produce numerical oscillations. Thus, we propose a downwind limiting method as a supplementary constraint of ENO property. The maximum value of θ which corresponds to the optimal accuracy of a weighting function can then be determined. The accuracy analysis via the standard Taylor expansion at critical points reveals that the proposed weighting functions outperform classical WENO schemes. The accuracy, scale-invariant property and ENO property of proposed schemes are further validated through benchmark tests, which shows the superior performance of the new schemes in comparison with existing three-cell-based WENO schemes. Thus, this work proposes a new direction to construct scale-invariant WENO weighting functions with optimal accuracy, which can potentially be extended to higher-order WENO schemes.