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. Sénéchal, and I. Vallet, J. Comput. Phys., 228 (2009), pp. 8481–8524] 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. 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.

In this work, a model for the simulation of infectious disease outbreaks including mobility data is presented. The model is based on the SAIR compartmental model and includes mobility data terms that model the flow of people between different regions. The aim of the model is to analyze the influence of mobility on the evolution of a disease after a lockdown period and to study the appearance of small epidemic outbreaks due to the so-called imported cases. We apply the model to the simulation of the COVID-19 in the various areas of Spain, for which the authorities made available mobility data based on the position of cell phones. We also introduce a method for the estimation of incomplete mobility data. Some numerical experiments show the importance of data completion and indicate that the model is able to qualitatively simulate the spread tendencies of small outbreaks. This work was motivated by an open call made to the mathematical community in Spain to help predict the spread of the epidemic.

A novel scheme, based on third-order weighted essentially nonoscillatory (WENO) reconstructions, is presented. It attains unconditionally optimal accuracy when the data is smooth enough, even in pr...