Improvements to enhance robustness of third-order scale-independent WENO-Z schemes

Abstract

Although there are many improvements to WENO3-Z that target the achievement of optimal order in the occurrence of the first-order critical point (CP1), they mainly address resolution performance, while the robustness of schemes is of less concern and lacks understanding accordingly. In light of our analysis considering the occurrence of critical points within grid intervals, we theoretically prove that it is impossible for a scale-independent scheme that has the stencil of WENO3-Z to fulfill the above order achievement, and current scale-dependent improvements barely fulfill the job when CP1 occurs at the middle of the grid cell. In order to achieve scale-independent improvements, we devise new smoothness indicators that increase the error order from 2 to 4 when CP1 occurs and perform more stably. Meanwhile, we construct a new global smoothness indicator that increases the error order from 4 to 5 similarly, through which new nonlinear weights with regard to WENO3-Z are derived and new scale-independents improvements, namely WENO-ZES2 and -ZES3, are acquired. Through 1D scalar and Euler tests, as well as 2D computations, in comparison with typical scale-dependent improvement, the following performances of the proposed schemes are demonstrated: The schemes can achieve third-order accuracy at CP1 no matter its location in the stencil, indicate high resolution in resolving flow subtleties, and manifest strong robustness in hypersonic simulations (e.g., the accomplishment of computations on hypersonic half-cylinder flow with Mach numbers reaching 16 and 19, respectively, as well as essentially non-oscillatory solutions of inviscid sharp double cone flow at M=9.59), which contrasts the comparative WENO3-Z improvement.