Improvements to Enhance Robustness of Third-Order Scale-Independent WENO-Z Schemes (Advances in Applied Mathematics and Mechanics, 2024)
Although there are many improvements to WENO3-Z with the target toachieve the optimal order in the occurrence of the first-order critical point ($CP_1$), theymainly address resolution performance, while the robustness of schemes is of less concern and lacks understanding accordingly. In light of our analysis considering theoccurrence 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 fulfillthe above order achievement, and current scale-dependent improvements barely fulfill the job when $CP_1$ occurs at the middle of the grid cell. In order to achieve scale-independent improvements, we devise new smoothness indicators that increase theerror order from 2 to 4 when $CP_1$ 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 andnew scale-independent improvements, namely WENO-${\rm Z}_{ES2}$ and -${\rm Z}_{ES3},$ are acquired.Through 1D scalar and Euler tests, as well as 2D computations, in comparison withtypical scale-dependent improvements, the following performances of the proposedschemes are demonstrated: the schemes can achieve third-order accuracy at $CP_1$ nomatter its location in the stencil, indicate high resolution in resolving flow subtleties,and manifest strong robustness in hypersonic simulations (e.g., the accomplishment ofcomputations on hypersonic half-cylinder flow with Mach numbers reaching 16 and19, respectively, as well as essentially non-oscillatory solutions of inviscid sharp double cone flow at $M=9.59$), which contrasts the comparative WENO3-Z improvements.