Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes

In the reconstruction step of (2r - 1) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights _αk and _ωk, such as the WENO-JS weights by Jiang et al. and the WENO-Z weights by Borges et al., are designed to recover the formal (2r - 1) order (optimal order) of the upwinded central finite difference scheme when the solution is sufficiently smooth. The smoothness of the solution is determined by the lower order local smoothness indicators β_k in each substencil. These nonlinear weight formulations share two important free parameters in common: the power p, which controls the amount of numerical dissipation, and the sensitivity ε, which is added to β_k to avoid a division by zero in the denominator of _αk. However, ε also plays a role affecting the order of accuracy of WENO schemes, especially in the presence of critical points. It was recently shown that, for any design order (2r - 1), ε should be of Ω(δ^x2) (Ω(δ^{x m}) means that ε ≥ Cδ^{x m} for some C independent of δx, as δx → 0) for the WENO-JS scheme to achieve the optimal order, regardless of critical points. In this paper, we derive an alternative proof of the sufficient condition using special properties of β_k. Moreover, it is unknown if the WENO-Z scheme should obey the same condition on ε. Here, using same special properties of β_k, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition ε = Ω(δ^{x m}), where m(r, p) ≥ 2 is the optimal sensitivity order, regardless of critical points. Both theoretical results are confirmed numerically on smooth functions with arbitrary order of critical points. This is a highly desirable feature, as illustrated with the Lax problem and the Mach 3 shock-density wave interaction of one dimensional Euler equations, for a smaller ε allows a better essentially non-oscillatory shock capturing as it does not over-dominate over the size of β_k. We also show that numerical oscillations can be further attenuated by increasing the power parameter 2 ≤ p ≤ r - 1, at the cost of increased numerical dissipation. Compact formulas ofβ_k for WENO schemes are also presented.