L∞L^\infty bounds for numerical solutions of noncoercive convection-diffusion equations

International audienceIn this work, we apply an iterative energy method à la de Giorgi in order to establish $L^\infty$ bounds for numerical solutions of noncoercive convection-diffusion equations with mixed Dirichlet-Neumann boundary conditions

Probabilistic analysis of the upwind scheme for transport

We provide a probabilistic analysis of the upwind scheme for multi-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we prove that the scheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for all a>0, for a Lipschitz continuous initial datum. Our analysis provides a new interpretation of the numerical diffusion phenomenon

Numerical schemes for semiconductors energy- transport models

International audienceWe introduce some finite volume schemes for unipolar energy-transportmodels. Using a reformulation in dual entropy variables, we can show the decay ofa discrete entropy with control of the discrete entropy dissipation

Uncertainty quantification for kinetic models in socio-economic and life sciences

Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of the modeling parameters. For this reason the presence of random inputs characterizing the parameters uncertainty should be considered as an essential feature in the modeling process. In this survey we introduce several examples of such kinetic models, that are mathematically described by nonlinear Vlasov and Fokker--Planck equations, and present different numerical approaches for uncertainty quantification which preserve the main features of the kinetic solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic Equations

State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD.: European Joint Programme on Radioactive Waste Management

Document information Project Acronym EURAD Project Title European Joint Programme on Radioactive Waste Management Project Type European Joint Programme (EJP) EC grant agreement No. 847593 Project starting / end date 1 st June 2019-30 May 2024 Work Package No. 4 Work Package Title Development and Improvement Of NUmerical methods and Tools for modelling coupled processes Work Package Acronym DONUT Deliverable No. 4.

An Approximate Two-Point Dirichlet Flux for Quasilinear Convection Diffusion Equations

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L∞ stability of the MUSCL methods

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