Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes
In the reconstruction step of (2r-1) ( 2 r - 1 ) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights ?k ? k and ?k ? 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 (2r-1) ( 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 ?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 ?, which is added to ?k to avoid a division by zero in the denominator of ?k ? k . However, ?  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) ( 2 r - 1 ) , ?  should be of ?(?x2) ? ( ? x 2 ) (?(?xm) ? ( ? x m ) means that ??C?xm ? ? C ? x m for some C  independent of ?x ? x , as ?x?0 ? 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 ?k. Moreover, it is unknown if the WENO-Z scheme should obey the same condition on ?. Here, using same special properties of ?k, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition ?=?(?xm) ? = ? ( ? x m ) , where m(r,p)?2 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 ? allows a better essentially non-oscillatory shock capturing as it does not over-dominate over the size of ?k. We also show that numerical oscillations can be further attenuated by increasing the power parameter 2?p?r-1 2 ? p ? r - 1 , at the cost of increased numerical dissipation. Compact formulas of?k for WENO schemes are also presented.
weighted essentially non-oscillatory nonlinear weights weno-js weno-z smoothness indicators power parameter hyperbolic equations sensitivity parameter