NDLI: Analysis of WENO Schemes for Full and Global Accuracy

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Mulet, P. Arndiga, F. Baeza, A. Belda, A. M.
Copyright Year	2011
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. 200212] to improve the order of accuracy of essentially nonoscillatory (ENO) reconstructions [A. Harten et al., J. Comput. Phys., 71 (1987), pp. 231303]. In [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202228], 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. 405452] and, later, [G. A. Gerolymos, D. Snchal, and I. Vallet, J. Comput. Phys., 228 (2009), pp. 84818524] obtained $(2r-1)$th order WENO reconstructions from rth 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. 542567], 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. 31913211; N. K. Yamaleev and M. H. Carpenter, J. Comput. Phys., 228 (2009), pp. 42484272] have addressed this problem with different weight designs for JiangShu smoothness indicators. In this paper we exploit the special structure of JiangShu 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.
Starting Page	893
Ending Page	915
Page Count	23
File Format	PDF
ISSN	00361429
DOI	10.1137/100791579
e-ISSN	10957170
Issue Number	2
Volume Number	49
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-04-27
Access Restriction	Subscribed
Subject Keyword	Stability and convergence of numerical methods smoothness indicators high resolution shock capturing schemes weighted essentially nonoscillatory nonlinear reconstructions Finite difference methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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