Antonio Baeza

A novel scheme, based on third-order Weighted Essentially Non-Oscillatory (WENO) reconstructions, is presented. It attains unconditionally optimal accuracy when the data is smooth enough, even in presence of critical points, and second-order accuracy if a discontinuity crosses the data. The key to attribute these properties to this scheme is the inclusion of an additional node in the data stencil, which is only used in the computation of the weights measuring the smoothness. The accuracy properties of this scheme are proven in detail and several numerical experiments are presented, which show that this scheme is more efficient in terms of the error reduction versus CPU time than its traditional third-order counterparts as well as several higher-order WENO schemes that are found in the literature.

A modified Weighted Essentially Non-Oscillatory (WENO) reconstruction technique preventing accuracy loss near critical points (regardless of their order) of the underlying data is presented. This approach only uses local data from the reconstruction stencil and does not rely on any sort of scaling parameters. The key novel ingredient is a weight design based on a new smoothness indicator, which defines the first WENO reconstruction procedure that never loses accuracy on smooth data, regardless of the presence of critical points of any order, and is therefore addressed as optimal WENO (OWENO) method. The corresponding weights are non-dimensional and scale-independent. The weight designs are supported by theoretical results concerning the accuracy of the smoothness indicators. The method is validated by numerical tests related to algebraic equations, scalar conservation laws, and systems of conservation laws.

In the context of cubic splines, the authors have contributed to a recent paper dealing with the computation of nonlinear derivatives at the interior nodes so that monotonicity is enforced while keeping the order of approximation of the spline as high as possible. During the review process of that paper, one of the reviewers raised the question of whether a cubic spline interpolating monotone data could be forced to preserve monotonicity by imposing suitable values of the first derivative at the endpoints. Albeit a negative answer appears to be intuitive, we have found no results regarding this fact. In this short work we prove that the answer to that question is actually negative.

Cost-effective multilevel techniques for homogeneous hyperbolic conservation laws are very successful in reducing the computational cost associated to high resolution shock capturing numerical schemes. Because they do not involve any special data structure, and do not induce savings in memory requirements, they are easily implemented on existing codes and are recommended for 1D and 2D simulations when intensive testing is required. The multilevel technique can also be applied to balance laws, but in this case, numerical errors may be induced by the technique. We present a series of numerical tests that point out that the use of monotonicity-preserving interpolatory techniques eliminates the numerical errors observed when using the usual 4-point centered Lagrange interpolation, and leads to a more robust multilevel code for balance laws, while maintaining the efficiency rates observed for hyperbolic conservation laws.

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.

In this paper we try to redesign an algorithm analyzed in previous works by Amat et al. in order to obtain a centered strategy aimed to approximate the solution of linear and nonlinear systems of conservation laws. We perform the analysis of the new algorithm in order to explain why the new strategy works close to shocks in the solution of conservation laws. The objective is to force the use of the most centered stencil of those used whenever the data are affected by the presence of a high gradient.

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...