Analysis of WENO Schemes for Full and Global Accuracy

Abstract

Liu, Osher, and Chan introduced weighted essentially nonoscillatory (WENO) reconstructions in [X.-D. Liu, S. Osher, and T. Chan, J. Comput. Phys., 115 (1994), pp. 200-212] to improve the order of accuracy of essentially nonoscillatory (ENO) reconstructions [A. Harten et al., J. Comput. Phys., 71 (1987), pp. 231-303]. In [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202-228], the authors proposed smoothness indicators to obtain a WENO fifth order reconstruction from third order ENO reconstructions. With these smoothness indicators, Balsara and Shu [J. Comput. Phys., 160 (2000), pp. 405-452] and, later, [G. A. Gerolymos, D. Senechal, and I. Vallet, J. Comput. Phys., 228 (2009), pp. 8481-8524] obtained $(2r-1)$th order WENO reconstructions from $r$th order ENO reconstructions for $4\leq r\leq 6$, resp., $7\leq r\leq 9$. In [A. K. Henrick, T. D. Aslam, and J. M. Powers, J. Comput. Phys., 207 (2005), pp. 542-567], the authors noticed that these reconstructions do not attain the optimal order $2r-1$ at extrema and they proposed a fix for the problem. Other authors [R. Borges et al., J. Comput. Phys., 227 (2008), pp. 3191-3211; N. K. Yamaleev and M. H. Carpenter, J. Comput. Phys., 228 (2009), pp. 4248-4272] have addressed this problem with different weight designs for Jiang-Shu smoothness indicators. In this paper we exploit the special structure of Jiang-Shu smoothness indicators and analyze the role of a parameter appearing in the weight definition to avoid division by zero to obtain for any $r\geq 2$, by standard approximation properties of Lagrange interpolation, that the order of the WENO reconstruction is $2r-1$ at smooth regions, regardless of neighboring extrema, whilst this order is $r$, as ENO reconstructions have, when the function has a discontinuity in the stencil of $2r-1$ points but it is smooth in at least one of the substencils of $r$ points. The optimal weights are also obtained in closed form.