Preventing numerical oscillations in the efficient component-wise WENO-ACM method for compressible flows with discontinuities

The component-wise WENO methods are very favorable for simulating complex problems without full characteristic structures in closed form, as they avoid the costly and dependency characteristic-decomposition process which is strictly required by their characteristic-wise counterparts. However, unfavorable numerical oscillations can be induced by the component-wise WENO methods, especially for those with very low dissipations. This study tries to overcome this crucial shortcoming existing in the component-wise WENO-ACM method, whose characteristic-wise counterpart was well-validated and proved to be highly efficient. The present analysis indicates that the numerical oscillations are basically due to the fact that the mapping function of WENO-ACM over-amplifies the contributions of less-smooth substencils. Therefore, a practical principle for designing the mapping function is proposed to prevent the numerical oscillations: the relative order of the nonlinear weights obtained from the classical WENO-JS method must be guaranteed. Obeying this principle, we present an effective but easy-implemented mapping function without over-amplifications to enhance the behavior of the component-wise WENO-ACM method in eliminating or significantly reducing the numerical oscillations with improved computational efficiency. Extensive numerical experiments governed by 1D and 2D Euler systems are performed to demonstrate the enhanced performances of the proposed method. We conclude that the proposed method can achieve the formal convergence rates of accuracy in regions with smooth solutions regardless of critical points and it is much more stable than the component-wise WENO-ACM method. Moreover, compared to the characteristic-wise WENO-ACM method, the new method can save 25–57% computational cost.