A robust and efficient component-wise WENO scheme for Euler equations

Component-wise WENO schemes are attractive to researchers for simulations of complex problems in practice especially for the cases whose full characteristic structures cannot be computed in closed form, as they avoid the expensive and complex characteristic-decomposition approach. However, these schemes suffer from spurious oscillations near discontinuities. In this work, we address this issue for the component-wise WENO-IM(2, 0.1) scheme whose characteristic-wise version has been well validated and proved to be very low dissipative in [H. Feng, C. Huang, R. Wang, An improved mapped weighted essentially non-oscillatory scheme, Appl. Math. Comput. 232 (2014) 453–468]. We propose to use a very simple modified mapping technique that removes or greatly decreases spurious oscillations for the component-wise WENO-IM(2, 0.1) scheme without causing any extra computational cost. Several benchmark tests modeled via one-dimensional and two-dimensional Euler equations are conducted to validate and evaluate the proposed scheme. The results demonstrate that the new scheme obtains the formal order of accuracy in smooth regions regardless of extreme points. Moreover, its robustness is as good as the characteristic-wise WENO-IM(2, 0.1) scheme. And consequently, the proposed scheme is much more stable than the component-wise WENO-IM(2, 0.1) scheme as expected. Besides, compared to the characteristic-wise WENO-IM(2, 0.1) scheme, the new method is about 30% to 50% faster in simulating two-dimensional problems, which indicates its excellent efficiency.