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SUMMARYThe numerical simulation of physical phenomena represented by non-linear hyperbolic systems of conservation laws presents speciÿc di culties mainly due to the presence of discontinuities in the solution. State of the art methods for the solution of such equations involve high resolution shock capturing schemes, which are able to produce sharp proÿles at the discontinuities and high accuracy in smooth regions, together with some kind of grid adaption, which reduces the computational cost by using ÿner grids near the discontinuities and coarser grids in smooth regions. The combination of both techniques presents intrinsic numerical and programming di culties. In this work we present a method obtained by the combination of a high-order shock capturing scheme, built from Shu-Osher's conservative formulation (J. Comput. Phys. 1988; 77:439-471; 83:32-78), a ÿfth-order weighted essentially non-oscillatory (WENO) interpolatory technique (J. Comput. Phys. 1996; 126:202-228) and Donat-Marquina's ux-splitting method (J. Comput. Phys. 1996; 125:42-58), with the adaptive mesh reÿnement (AMR) technique of Berger and collaborators (Adaptive mesh reÿnement for hyperbolic partial di erential equations.

A novel central weighted essentially non-oscillatory (central WENO; CWENO)-type scheme for the construction of high-resolution approximations to discontinuous solutions to hyperbolic systems of conservation laws is presented. This procedure is based on the construction of a global average weight using the whole set of Jiang-Shu smoothness indicators associated to every candidate stencil. By this device one does not to have to rely on ideal weights, which, under certain stencil arrangements and interpolating point locations, do not define a convex combination of the lower-degree interpolating polynomials of the corresponding substencils. Moreover, this procedure also prevents some cases of accuracy loss near smooth extrema that are experienced by classical WENO and CWENO schemes. These properties result in a more flexible scheme that overcomes these issues, at the cost of only a few additional computations with respect to classical WENO schemes and with a smaller cost than classical CWENO schemes. Numerical examples illustrate that the proposed CWENO schemes outperform both the traditional WENO and the original CWENO schemes.

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.

Optimal aerodynamic shape design aims to find the minimum of a functional that describes an aerodynamic property, by controlling the partial differential equation modeling the dynamics of the flow that surrounds an aircraft, by using surface deformation techniques. As a solution to the enormous computational resources required for classical shape optimization of functionals of aerodynamic interest, probably the best strategy is to apply methods inspired in control theory. One of the key ingredients relies on the usage of the adjoint methodology to simplify the computation of gradients. In this paper we restrict our attention to optimal shape design in two-dimensional systems governed by the steady Euler equations for flows whose steady-state solutions present discontinuities in the flow variables (an isolated shock wave). We first review some facts on control theory applied to optimal shape design and recall the 2-D Euler equations (including the Rankine-Hugoniot conditions). We then study the adjoint formulation, providing a detailed exposition of how the derivatives of functionals of aeronautical interest may be obtained when a discontinuity appears. Further on, adjoint equations will be discretized and analyzed and some novel numerical experiments with adjoint Rankine-Hugoniot relations will be shown. Finally, we expose some conclusions about the viability of a rigorous approach to the continuous Euler adjoint system with discontinuities in the flow variables. Nomenclature A = Jacobian matrix for the convective fluxes C D = drag coefficient C L = lift coefficient C p = pressure coefficient c = local speed of sound E = total energy F = vector of convective fluxes f = vector of numerical fluxes H = enthalpy J = cost function n = normal vector P = static pressure S = solid wall boundary S ad = space of admissible surfaces s = speed of shock wave propagation t = unit tangent vector U = vector of conserved variables v = velocity vector W = vector of characteristic variables x = Cartesian coordinates vector x b = intersection between a shock wave and a solid surface = angle of attack 1= "far-field" boundary = ratio of specific heat = first difference = first variation @ = partial derivative @ n = normal derivative to a curve @ tg = tangent derivative to a curve = curve parameter = curvature of a curve = diagonal matrix of inviscid eigenvalues = density = shock wave curve = vector of adjoint variables = fluid domain

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The application of suitable numerical boundary conditions for hyperbolic conservation laws on domains with complex geometry has become a problem with certain difficulty that has been tackled in different ways according to the nature of the numerical methods and mesh type. In this paper we present a technique for the extrapolation of information from the interior of the computational domain to ghost cells designed for structured Cartesian meshes (which, as opposed to non-structured meshes, cannot be adapted to the morphology of the domain boundary). This technique is based on the application of Lagrange interpolation with a filter for the detection of discontinuities that permits a data dependent extrapolation, with higher order at smooth regions and essentially non oscillatory properties near discontinuities.