Discrete energy estimates for the abcd-systems

International audienceIn this article, we propose finite volume schemes for the abcd-systems and we establish stability and error estimates. The order of accuracy depends on the so-called BBM-type dispersion coefficients b and d. If bd > 0, the numerical schemes are O(∆t + (∆x)^2) accurate, while if bd = 0, we obtain an O(∆t + ∆x)-order of convergence. The analysis covers a broad range of the parameters a,b,c,d. In the second part of the paper, numerical experiments validating the theoretical results as well as head-on collision of traveling waves are investigated

Linear stability of a vectorial kinetic relaxation scheme with a central velocity

International audienceThis article deals with the linear stability of an implicit vectorial kinetic relaxation scheme with a central velocity used to solve numerically some multi-scale hyperbolic systems

Linear stability of a vectorial kinetic relaxation scheme with a central velocity

International audienceThis article deals with the linear stability of an implicit vectorial kinetic relaxation scheme with a central velocity used to solve numerically some multi-scale hyperbolic systems

Error estimates of finite difference schemes for the Korteweg-de Vries equation

International audienceThis article deals with the numerical analysis of the Cauchy problem for the Korteweg-de Vries equation with a finite difference scheme. We consider the Rusanov scheme for the hyperbolic flux term and a 4-points $\theta$-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant-Friedrichs-Lewy condition when $\theta \geq \frac{1}{2}$ and under an "Airy" Courant-Friedrichs-Lewy condition when $\theta<\frac{1}{2}$. More precisely, we get the first order convergence rate for strong solutions in the Sobolev space $H^s(\mathbb{R})$, $s \geq 6$ and extend this result to the non-smooth case for initial data in $H^s(\mathbb{R})$, with $s\geq \frac{3}{4}$, to the price of a loss in the convergence order. Numerical simulations indicate that the orders of convergence may be optimal when $s\geq3$

Vectorial kinetic relaxation model with central velocity. Application to implicit relaxations schemes

International audienceWe apply flux vector splitting (FVS) strategy to the implicit kinetic schemes for hyperbolic systems. It enables to increase the accuracy of the method compared to classical kinetic schemes while still using large time steps compared to the characteristic speeds of the problem. The method also allows to tackle multi-scale problems, such asthelow Machnumber limit, for which wavespeeds with larger atioare involved. We present several possible kinetic relaxation schemes based on FVS and compare them on one-dimensional test-cases. We discuss stability issues for this kind of metho

Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples

In this paper, we consider a ferromagnetic material of ellipsoidal shape. The associated magnetic moment then has two asymptotically stable opposite equilibria, of the form $\pm\overline{m}$. In order to use these materials for memory storage purposes, it is necessary to know how to control the magnetic moment. We use as a control variable a spatially uniform external magnetic field and consider the question of flipping the magnetic moment, i.e., changing it from the $+\overline{m}$ configuration to the $-\overline{m}$ one, in minimal time. Of course, it is necessary to impose restrictions on the external magnetic field used. We therefore include a constraint on the $L^\infty$ norm of the controls, assumed to be less than a threshold value $U$. We show that, generically with respect to the dimensions of the ellipsoid, there is a minimal value of $U$ for this problem to have a solution. We then characterize it precisely. Finally, we investigate some particular configurations associated to geometries enjoying symmetries properties and show that in this case the magnetic moment can be controlled in minimal time without imposing a threshold condition on $U$

Study of Physics-Based preconditioning with High-order Galerkin discretization for hyperbolic wave problems

In this article, we detail the construction of a physics-based preconditioner. The Schur decomposition is the key point of the method which is tested on two hyperbolic systems : acoustic wave equations and shallow water equations without source term. Some conserved properties between preconditoner and initial operator are discussed, especially the propagation speeds of a plane wave

Study of Physics-Based preconditioning with High-order Galerkin discretization for hyperbolic wave problems

In this article, we detail the construction of a physics-based preconditioner. The Schur decomposition is the key point of the method which is tested on two hyperbolic systems : acoustic wave equations and shallow water equations without source term. Some conserved properties between preconditoner and initial operator are discussed, especially the propagation speeds of a plane wave

Reduced modelling and optimal control of epidemiological individual-based models with contact heterogeneity

Modelling epidemics via classical population-based models suffers from shortcomings that socalled individual-based models are able to overcome, as they are able to take heterogeneity features into account, such as super-spreaders, and describe the dynamics involved in small clusters. In return, such models often involve large graphs which are expensive to simulate and difficult to optimize, both in theory and in practice. By combining the reinforcement learning philosophy with reduced models, we propose a numerical approach to determine optimal health policies for a stochastic epidemiological graphmodel taking into account super-spreaders. More precisely, we introduce a deterministic reduced population-based model involving a neural network, and use it to derive optimal health policies through an optimal control approach. It is meant to faithfully mimic the local dynamics of the original, more complex, graph-model. Roughly speaking, this is achieved by sequentially training the network until an optimal control strategy for the corresponding reduced model manages to equally well contain the epidemic when simulated on the graph-model. After describing the practical implementation of this approach, we will discuss the range of applicability of the reduced model and to what extent the estimated control strategies could provide useful qualitative information to health authorities