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

• Authors: Wai-Sun Don;Rafael Borges
• Venue: Journal of Computational Physics
• Year: 2013

In the reconstruction step of (2r-1) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights @a"k and @w"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) 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 @b"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 @e, which is added to @b"k to avoid a division by zero in the denominator of @a"k. However, @e 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), @e should be of @W(@Dx^2) (@W(@Dx^m) means that @e=C@Dx^m for some C independent of @Dx, as @Dx-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 @b"k. Moreover, it is unknown if the WENO-Z scheme should obey the same condition on @e. Here, using same special properties of @b"k, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition @e=@W(@Dx^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 @e allows a better essentially non-oscillatory shock capturing as it does not over-dominate over the size of @b"k. We also show that numerical oscillations can be further attenuated by increasing the power parameter 2=