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Emmanuel Frenod

Publications and source records attributed to Emmanuel Frenod.

At least 19 recordsLinked to original sources
arXiv2016

Integral operator approach over octonions to solution of nonlinear PDE

Integration of nonlinear partial differential equations with the help of the non-commutative integration over octonions is studied. An apparatus permitting to take into account symmetry properties of PDOs is developed. For this purpose formulas for calculations of commutators of integral and partial differential operators are deduced. Transformations of partial differential operators and solutions of partial differential equations are investigated. Theorems providing solutions of nonlinear PDEs are proved. Examples are given. Applications to PDEs of hydrodynamics and other types PDEs are described.

math.AP↗
arXiv2013

Modeling and Solving Alternative Financial Solutions Seeking

In this paper we build a method to optimize Multi-Year Prospective Budgets. First we present a systemic model of Local Community Finances. Then, from two acceptable Multi-Year Prospective Budgets the method implements a Genetic Algorithm to generate a collection of admissible Multi-Year Prospective Budgets among which Decision-Makers can choose. The method is tested on simplified cases and on in operational situation and gives satisfactory results.

q-fin.GN↗
arXiv2013

A PDE-like Toy-Model of Territory Working

This note introduces a PDE-like Toy-Model that embeds several aspects of Territory Working. The long term goal is to build a software tool that behaves like a Territory, and in particular that incorporates its multi-scale-in-time-and-space nature, in order to make simulations, to explore scenarios, to foresee policy impacts, to help to make a decision when facing a change in the environment or in cultural behavior, {\it etc.}. The aim of this note is prove that the concept of building a model that couple Systemic Approach and PDE tools to achieve the evoked long term goal is possible.

math.AP↗
arXiv2013

Two-Scale numerical simulation of sand transport problems

In this paper we consider a model for short term dynamics of dunes in tidal area. We construct a Two-Scale Numerical Method based on the fact that the solution of the equation which has oscillations Two-Scale converges to the solution of a well-posed problem. This numerical method uses on Fourier series.

math.NA↗
arXiv2013

Derivation of the Isotropic Diffusion Source Approximation (IDSA) for Supernova Neutrino Transport by Asymptotic Expansions

We present Chapman--Enskog and Hilbert expansions applied to the $\BigO(v/c)$ Boltzmann equation for the radiative transfer of neutrinos in core-collapse supernovae. Based on the Legendre expansion of the scattering kernel for the collision integral truncated after the second term, we derive the diffusion limit for the Boltzmann equation by truncation of Chapman--Enskog or Hilbert expansions with reaction and collision scaling. We also give asymptotically sharp results obtained by the use of an additional time scaling. The diffusion limit determines the diffusion source in the \emph{Isotropic Diffusion Source Approximation (IDSA)} of Boltzmann's equation for which the free streaming limit and the reaction limit serve as limiters. Here, we derive the reaction limit as well as the free streaming limit by truncation of Chapman--Enskog or Hilbert expansions using reaction and collision scaling as well as time scaling, respectively. Finally, we motivate why limiters are a good choice for the definition of the source term in the IDSA.

math.AP↗
arXiv2013

An exponential integrator for a highly oscillatory Vlasov equation

In the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an exponential time differencing approach, the scheme is able to use large time steps with respect to the typical size of the fast oscillations of the solution.

math.NA↗
arXiv2013

Paralic confinement computations in coastal environment with interlocked areas

This paper is in the continuity of a work program. Its goal is to develop an approach of the paralic confinement usable from the modeling slant, before implementing it in numerical tools. More specifically, we here deal with the multiscale aspect of the confinement. If a paralic environment is separated into two (or more) connected areas, we will show that is possible to split the confinement problem into two related problems, one for each area. At the end of this paper, we will focus on the importance of the interface length between the two subdomains.

math.NA↗
arXiv2012

A Mathematical Description of the IDSA for Supernova Neutrino transport, its discretization and a comparison with a finite volume scheme for Boltzmann's Equation

In this paper we give an introduction to the Boltzmann equation for neutrino transport used in core collapse supernova models as well as a detailed mathematical description of the \emph{Isotropic Diffusion Source Approximation} (IDSA). Furthermore, we present a numerical treatment of a reduced Boltzmann model problem based on time splitting and finite volumes and revise the discretization of the IDSA for this problem. Discretization error studies carried out on the reduced Boltzmann model problem and on the IDSA show that the errors are of order one in both cases. By a numerical example, a detailed comparison of the reduced model and the IDSA is carried out and interpreted. For this example the IDSA modeling error with respect to the reduced Boltzmann model is numerically determined and localized.

astro-ph.SR↗
arXiv2012

Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field

In this paper, we build a Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field. This consists in writing the solution of this equation as a sum of two oscillating functions with circonscribed oscillations. The first of these functions has a shape which is close to the shape of the Two-Scale limit of the solution and the second one is a correction built to offset this imposed shape. The aim of such a decomposition is to be the starting point for the construction of Two-Scale Asymptotic-Preserving Schemes.

math.FA↗
arXiv2011

The Gyro-Kinetic Approximation. An attempt at explaining the method based on Darboux Algorithm and Lie Transform

This Proceeding presents the method that allows us to get the Gyro-Kinetic Approximation of the Dynamical System satisfied by the trajectory of a particle submitted to a Strong Magnetic Field. The goal of the method is to build a change of coordinates in order to make the dynamic of two components of the trajectory to disappear. This change of coordinates is based on a Darboux mathematical Algorithm and on a Lie Transform

math.FA↗
arXiv2011

Long term behaviour of singularly perturbed parabolic degenerated equation

In this paper we consider models for short-term, mean-term and long-term morphodynamics of dunes and megariples. We give an existence and uniqueness result for long term dynamics of dunes. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the mean-term and long-term models are homogenized.

math.AP↗
arXiv2010

Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates

In this paper, we present some new results about the approximation of the Vlasov-Poisson system with a strong external magnetic field by the 2D finite Larmor radius model. The proofs within the present work are built by using two-scale convergence tools, and can be viewed as a new slant on previous works of Fr\'enod and Sonnendr\"ucker and Bostan on the 2D finite Larmor Radius model. In a first part, we recall the physical and mathematical contexts. We also recall two main results from previous papers of Fr\'enod and Sonnendr\"ucker and Bostan. Then, we introduce a set of variables which are so-called canonical gyrokinetic coordinates, and we write the Vlasov equation in these new variables. Then, we establish some two-scale convergence and weak-* convergence results.

math.NA↗
arXiv2010

Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment

In this paper we build models for short-term, mean-term and long-term dynamics of dune and megariple morphodynamics. They are models that are degenerated parabolic equations which are, moreover, singularly perturbed. We, then give an existence and uniqueness result for the short-term and mean-term models. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the short-term model is homogenized.

math.AP↗
arXiv2009

Models for cohesive sediments describing the evolution of the characteristics of particles

The goal of this paper is to set up a framework designed to take into account the characteristics of sediment particles when transported by water. Our protocol consists in describing the characteristics of sediment particles via an additional variable, and to build operators involving this new variable, modeling the evolution of the particle characteristics. Several such operators are proposed, some based on principles of relaxation toward an equilibrium, and others on a description of the particles' aggregation and fragmentation process. A discrete version of the latter is also offered for numerical settings.

math.AP↗
arXiv2008

An explanatory model to validate the way water activity rules periodic terrace generation in Proteus mirabilis swarm

In this paper we explain the biophysical principles which, according to us, govern the Proteus mirabilis swarm phenomenon. Then, we translate this explanation into a mathematical model, essentially based on partial differential equations. This model is then implemented using numerical methods of the finite volume type in order to make simulations. The simulations show most of the characteristics which are observed in situ and in particular the terrace generation.

math.AP↗
arXiv2008

Modeling the Coastal Ocean over a Time Period of Several Weeks

From a scale analysis of hydrodynamic phenomena having a significant action on the drift of an object in coastal ocean waters, we deduce equations modeling the associated hydrodynamic fields over a time period of several weeks. These models are essentially non linear hyperbolic systems of PDE involving a small parameter. Then from the models we extract a simplified and nevertheless typical one for which we prove that its classical solution exists on a time interval which is independent of the small parameter. We then show that the solution weak-* converges as the small parameter goes to zero and we characterize the equation satisfied by the weak-* limit

math.AP↗