1,721,011 research outputs found
High order central WENO-Implicit-Explicit Runge Kutta schemes for the BGK model on general polygonal meshes
No full textIn this work, a family of high order accurate Central Weighted ENO (CWENO) finite volume schemes for the solution of nonlinear kinetic equation of relaxation type is presented. After discretization of the velocity space by using a discrete ordinate approach, the space reconstruction is realized by integration over conformal arbitrary shaped closed space control volumes in a CWENO fashion. Compared to other WENO methods on unstructured meshes, in the method here presented, the total stencil size is the minimum possible and the linear weights can be arbitrarily chosen. These two aspects make their use for kinetic equations and the practical implementation on general unstructured meshes particularly interesting. The full discretization is then obtained by combining the previous phase-space approximation with an Implicit-Explicit Runge Kutta high order time discretization which guarantees stability, accuracy and preservation of the asymptotic state. In particular, to guarantee in the finite volume framework space accuracy higher than two, a new class of IMEX methods has been set into place and its properties have been studied. The formal order of accuracy is numerically measured for different regimes, computational performances of the proposed class of methods are tested on several standard two dimensional benchmark problems for kinetic equations. The novel methods are finally applied to a prototype engineering problem consisting in a supersonic flow around a NACA 0012 airfoil. In our computations we employ up to ≈325 millions of degrees of freedom and 256 GB of RAM run on 128 cores with Fortran-MPI providing evidence that the above schemes are suitable for implementation on parallel distributed memory supercomputers
The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations
No full textWe study the derivation of second order macroscopic traffic models from kinetic descriptions. In particular, we recover the celebrated Aw–Rascle model as the hydrodynamic limit of an Enskog-type kinetic equation out of a precise characterisation of the microscopic binary interactions among the vehicles. Unlike other derivations available in the literature, our approach unveils the multiscale physics behind the Aw–Rascle model. This further allows us to generalise it to a new class of second order macroscopic models complying with the Aw–Rascle consistency condition, namely the fact that no wave should travel faster than the mean traffic flow
High order modal Discontinuous Galerkin Implicit–Explicit Runge Kutta and Linear Multistep schemes for the Boltzmann model on general polygonal meshes
No full textDeterministic solutions of the Boltzmann equation represent a real challenge due to the enormous computational effort which is required to produce such simulations and often stochastic methods such as Direct Simulation Monte Carlo (DSMC) are used instead due to their lower computational cost. In this work, we show that combining different technologies for the discretization of the velocity space and of the physical space coupled with suitable time integration techniques, it is possible to compute very precise deterministic approximate solutions of the Boltzmann model in different regimes, from extremely rarefied to dense fluids, with CFL conditions only driven by the hyperbolic transport term. To that aim, we develop modal Discontinuous Galerkin (DG) Implicit–Explicit Runge Kutta schemes (DG-IMEX-RK) and Implicit–Explicit Linear Multistep Methods based on Backward-Finite-Differences (DG-IMEX-BDF) for solving the Boltzmann model on multidimensional unstructured meshes. The solution of the Boltzmann collision operator is obtained through fast spectral methods, while the transport term in the governing equations is discretized relying on an explicit shock-capturing DG method on polygonal tessellations in the physical space. A novel class of WENO-type limiters, based on a shifting of the moments of inertia for each zone of the mesh, is used to control spurious oscillations of the DG solution across discontinuities. The use of Linear Multistep Methods (LMM) allows the Boltzmann solutions to be consistent not only with the compressible Euler limit but also with the Navier–Stokes asymptotic regime. In addition, as numerically proven, they also permit to strongly reduce the computational effort compared to Runge–Kutta approaches while maintaining the same or even larger accuracy. The performances of these different time discretization techniques are measured comparing both precision and efficiency. At the same time, comparisons against simpler relaxation type kinetic models such as the BGK model are proposed. The order of convergence is numerically measured for different regimes and found to agree with the theoretical findings. The new methods are validated considering two-dimensional benchmark test cases typically used in the fluid dynamics community. A prototype engineering problem consisting of a supersonic flow around a NACA 0012 airfoil with space–time-dependent boundary conditions is also presented for which the pressure coefficients are measured
Social climbing and Amoroso distribution
No full textWe introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker-Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker-Planck type equation with variable coefficients of diffusion and drift, characterized by the mathematical properties of the value function. The steady states of the statistical distribution of the social status predicted by the Fokker-Planck equations belong to the class of Amoroso distributions with Pareto tails, which correspond to the emergence of a social elite. The details of the microscopic kinetic interaction allow to clarify the meaning of the various parameters characterizing the resulting equilibrium. Numerical results then show that the steady state of the underlying kinetic equation is close to Amoroso distribution even in an intermediate regime in which interactions are not grazing
High order finite volume schemes with IMEX time stepping for the Boltzmann model on unstructured meshes
No full textIn this work, we present a family of time and space high order finite volume schemes for the solution of the full Boltzmann equation. The velocity space is approximated by using a discrete ordinate approach while the collisional integral is approximated by spectral methods. The space reconstruction is implemented by integrating the distribution function, which describes the state of the system, over arbitrarily shaped and closed control volumes using a Central Weighted ENO (CWENO) technique. Compared to other reconstruction methods, this approach permits to keep compact stencil sizes which is a remarkable property in the context of kinetic equations due to the considerable demand of computational resources. The full discretization is then obtained by combining the previous phase-space approximation with high order Implicit–Explicit (IMEX) Runge–Kutta schemes. These methods guarantee stability, accuracy and preservation of the asymptotic state. Comparisons of the Boltzmann model with simpler relaxation type kinetic models (like BGK) are proposed showing the capability of the Boltzmann equation to capture different physical solutions. The theoretical order of convergence is numerically measured in different regimes and the methods are tested on several standard two-dimensional benchmark problems in comparison with Direct Simulation Monte Carlo results. The article ends with a prototype engineering problem consisting of a subsonic and a supersonic flow around a NACA 0012 airfoil. All test cases are run with MPI parallelization on several cores, thus making the proposed methods suitable for parallel distributed memory supercomputers
A multi-agent description of the influence of higher education on social stratification
No full textWe introduce and discuss a system of one-dimensional kinetic equations describing the influence of higher education in the social stratification of a multi-agent society. The system is obtained by coupling a model for knowledge formation with a kinetic description of the social climbing in which the parameters characterizing the elementary interactions leading to the formation of a social elite are assumed to depend on the degree of knowledge/education of the agents. In addition, we discuss the case in which the education level of an individual is function of the position occupied in the social ranking. With this last assumption, we obtain a fully coupled model in which knowledge and social status influence each other. In the last part, we provide several numerical experiments highlighting the role of education in reducing social inequalities and in promoting social mobility
Optimal control of epidemic spreading in presence of social heterogeneity
Full text linkThe spread of COVID-19 has been thwarted in most countries through
non-pharmaceutical interventions. In particular, the most effective measures in
this direction have been the stay-at-home and closure strategies of businesses
and schools. However, population-wide lockdowns are far from being optimal
carrying heavy economic consequences. Therefore, there is nowadays a strong
interest in designing more efficient restrictions. In this work, starting from
a recent kinetic-type model which takes into account the heterogeneity
described by the social contact of individuals, we analyze the effects of
introducing an optimal control strategy into the system, to limit selectively
the mean number of contacts and reduce consequently the number of infected
cases. Thanks to a data-driven approach, we show that this new mathematical
model permits to assess the effects of the social limitations. Finally, using
the model introduced here and starting from the available data, we show the
effectivity of the proposed selective measures to dampen the epidemic trends
A data-driven kinetic model for opinion dynamics with social network contacts
No full textOpinion dynamics is an important and very active area of research that delves into the complex processes through which individuals form and modify their opinions within a social context. The ability to comprehend and unravel the mechanisms that drive opinion formation is of great significance for predicting a wide range of social phenomena such as political polarisation, the diffusion of misinformation, the formation of public consensus and the emergence of collective behaviours. In this paper, we aim to contribute to that field by introducing a novel mathematical model that specifically accounts for the influence of social media networks on opinion dynamics. With the rise of platforms such as Twitter, Facebook, and Instagram and many others, social networks have become significant arenas where opinions are shared, discussed and potentially altered. To this aim after an analytical construction of our new model and through incorporation of real-life data from Twitter, we calibrate the model parameters to accurately reflect the dynamics that unfold in social media, showing in particular the role played by the so-called influencers in driving individual opinions towards predetermined directions
An efficient second order all Mach finite volume solver for the compressible Navier–Stokes equations
No full textIn the numerical simulation of fluid dynamic problems there are situations in which acoustic waves are very fast compared to the average velocity of the fluid and conversely situations in which the fluid moves at high speed and shock waves may be present. Ideally, a numerical method should be able to treat these different regimes without strong limitations in terms of time step and without excessive related computational cost. Unfortunately, standard explicit in time schemes often adopted for hyperbolic problems are not suitable for these problems, hence remedies have to be studied. To this aim, the results presented in this article concern the development of a second order in time and space numerical method for the compressible Navier–Stokes equation which works for both high and low Mach numbers. In particular, when the Mach number goes to zero, one recovers a numerical method for the limit Navier–Stokes system which under some additional hypothesis degenerates to the incompressible Navier–Stokes equations, while in the case of high Mach numbers the method exhibits a shock capturing structure. The idea is based on partitioning the equations into a fast and a slow scale and by taking implicit the fast scale dynamic together with the viscous terms. The resulting numerical scheme is stable for time steps which are independent both from the speed of the pressure waves and from the diffusive terms characterizing the viscous forces and the heat flux. The only time step limitation is induced by the average speed of the flow. The work here presented extends the seminal ideas developed in Dimarco et al. (2017, 2018) for isentropic Euler equations and in Boscheri et al. (2020) for the full set of compressible Euler equations to the multidimensional Navier–Stokes system and permits efficient three dimensional simulations of all Mach problems. The discretization is constructed on Cartesian meshes and the method is second order accurate in space and time. Numerical results show the accuracy, the robustness and the effectiveness of the new proposed approach
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