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

In the reconstruction step of ( 2 r - 1 ) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights α k and ω k , 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 ( 2 r - 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 α k . 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 ( 2 r - 1 ) , ε should be of Ω ( Δ x 2 ) ( Ω ( Δ x m ) means that ε ⩾ C Δ x m for some C independent of Δ x , as Δ x → 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 of β k for WENO schemes are also presented.