arXiv subjects

Erwan Faou

Publications and source records attributed to Erwan Faou.

At least 37 records · Page 2Linked to original sources

arXiv2017

Free Vibrations of Axisymmetric Shells: Parabolic and Elliptic cases

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{é} system) are determined by an asymptotic analysis as the thickness ($2\varepsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency $k$ and the half-thickness $\varepsilon$. Optimizing $k$ for each chosen $\varepsilon$, we find power laws for $k$ in function of $\varepsilon$ that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{é} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{é} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\varepsilon$ tends to $0$. Its angular frequency exhibits a power law relationof the form $k=γ\varepsilon^{-β}$ with $β=\frac14$ in the parabolic case (cylinders and trimmed cones), and the various $β$s $\frac25$, $\frac37$, and $\frac13$ in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented.

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arXiv2016

An asymptotic preserving scheme for the relativistic Vlasov--Maxwell equations in the classical limit

We consider the relativistic Vlasov--Maxwell (RVM) equations in the limit when the light velocity $c$ goes to infinity. In this regime, the RVM system converges towards the Vlasov--Poisson system and the aim of this paper is to construct asymptotic preserving numerical schemes that are robust with respect to this limit. Our approach relies on a time splitting approach for the RVM system employing an implicit time integrator for Maxwell's equations in order to damp the higher and higher frequencies present in the numerical solution. A number of numerical simulations are conducted in order to investigate the performances of our numerical scheme both in the relativistic as well as in the classical limit regime. In addition, we derive the dispersion relation of the Weibel instability for the continuous and the discretized problem.

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arXiv2016

Convergence of a Normalized Gradient Algorithm for Computing Ground States

We consider the approximation of the ground state of the one-dimensional cubic nonlinear Schr{ö}dinger equation by a normalized gradient algorithm combined with linearly implicit time integrator, and finite difference space approximation. We show that this method, also called imaginary time evolution method in the physics literature, is con-vergent, and we provide error estimates: the algorithm converges exponentially towards a modified solitons that is a space discretization of the exact soliton, with error estimates depending on the discretization parameters.

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arXiv2016

High frequency oscillations of first eigenmodes in axisymmetric shells as the thickness tends to zero

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2$ε$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue $λ$($ε$) and the eigenvector angular frequency k($ε$) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts. First, the scalar Laplace operator for acoustics is addressed, for which k($ε$) is always zero. In contrast to it, for the Lam{é} system of linear elasticity several different types of shells are defined, characterized by their geometry, for which k($ε$) tends to infinity as $ε$ tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide $λ$($ε$) and k($ε$) and demonstrate by numerical examples the different behavior as $ε$ tends to zero.

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arXiv2015

On numerical Landau damping for splitting methods applied to the Vlasov-HMF model

We consider time discretizations of the Vlasov-HMF (Hamiltonian Mean-Field) equation based on splitting methods between the linear and non-linear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time stepsize.

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arXiv2015

High-order Hamiltonian splitting for Vlasov-Poisson equations

We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions, we study the order conditions of such methods. It appears that these conditions are of Runge-Kutta-Nystr{ö}m type. In the one dimensional case, the order conditions can be further simplified, and efficient methods of order 6 with a reduced number of stages can be constructed. In the general case, high-order methods can also be constructed using explicit computations of commutators. Numerical results are performed and show the benefit of using high-order splitting schemes in that context. Complete and self-contained proofs of convergence results and rigorous error estimates are also given.

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arXiv2015

Collision of almost parallel vortex filaments

We investigate the occurrence of collisions in the evolution of vortex filaments through a system introduced by Klein, Majda and Damodaran [KMD95] and Zakharov [Z88,Z99]. We first establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. The collision mechanism is based on the one of the self-similar solutions of the model, described in [BFM14]. In the second part of this paper we extend this construction to the case of an arbitrary number of filaments, with polygonial symmetry, that are perturbations of a configuration of parallel vortex filaments forming a polygon, with or without its center, rotating with constant angular velocity.

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arXiv2014

Simulations of Kinetic Electrostatic Electron Nonlinear (KEEN) Waves with Variable Velocity Resolution Grids and High-Order Time-Splitting

KEEN waves are nonlinear, non-stationary, self-organized asymptotic states in Vlasov plasmas outside the scope or purview of linear theory constructs such as electron plasma waves or ion acoustic waves. Nonlinear stationary mode theories such as those leading to BGK modes also do not apply. The range in velocity that is strongly perturbed by KEEN waves depends on the amplitude and duration of the ponderomotive force used to drive them. Smaller amplitude drives create highly localized structures attempting to coalesce into KEEN waves. These cases have much more chaotic and intricate time histories than strongly driven ones. The narrow range in which one must maintain adequate velocity resolution in the weakly driven cases challenges xed grid numerical schemes. What is missing there is the capability of resolving locally in velocity while maintaining a coarse grid outside the highly perturbed region of phase space. We here report on a new Semi-Lagrangian Vlasov-Poisson solver based on conservative non-uniform cubic splines in velocity that tackles this problem head on. An additional feature of our approach is the use of a new high-order time-splitting scheme which allows much longer simulations per computational e ort. This is needed for low amplitude runs which take a long time to set up KEEN waves, if they are able to do so at all. The new code's performance is compared to uniform grid simulations and the advantages quanti ed. The birth pains associated with KEEN waves which are weakly driven is captured in these simulations. These techniques allow the e cient simulation of KEEN waves in multiple dimensions which will be tackled next as well as generalizations to Vlasov-Maxwell codes which are essential to understanding the impact of KEEN waves in practice.

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arXiv2014

Landau damping in Sobolev spaces for the Vlasov-HMF model

We consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping.

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arXiv2014

Collisions of vortex filament pairs

We consider the problem of collisions of vortex filaments for a model introduced by Klein, Majda and Damodaran, and Zakharov to describe the interaction of almost parallel vortex filaments in three-dimensional fluids. Since the results of Crow examples of collisions are searched as perturbations of antiparallel translating pairs of filaments, with initial perturbations related to the unstable mode of the linearized problem; most results are numerical calculations. In this article we first consider a related model for the evolution of pairs of filaments and we display another type of initial perturbation leading to collision in finite time. Moreover we give numerical evidence that it also leads to collision through the initial model. We finally study the self-similar solutions of the model.

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arXiv2014

A Hamiltonian splitting for the Vlasov-Maxwell system

A new splitting is proposed for solving the Vlasov-Maxwell system. This splitting is based on a decomposition of the Hamiltonian of the Vlasov-Maxwell system and allows for the construction of arbitrary high order methods by composition (independent of the specific deterministic method used for the discretization of the phase space). Moreover, we show that for a spectral method in space this scheme satisfies Poisson's equation without explicitly solving it. Finally, we present some examples in the context of the time evolution of an electromagnetic plasma instability which emphasizes the excellent behavior of the new splitting compared to methods from the literature.

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arXiv2013

Fast weak-KAM integrators for separable Hamiltonian systems

We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a geometric integrator satisfying a discrete weak-KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min-plus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.

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arXiv2013

Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation

Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a non-resonance condition. They can both be verified if the time step-size is restricted by a CFL condition in the case of a constant plane wave. The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.

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arXiv2013

Resonant time steps and instabilities in the numerical integration of Schrödinger equations

We consider the linear and non linear cubic Schrödinger equations with periodic boundary conditions, and their approximations by splitting methods. We prove that for a dense set of arbitrary small time steps, there exists numerical solutions leading to strong numerical instabilities preventing the energy conservation and regularity bounds obtained for the exact solution. We analyze rigorously these instabilities in the semi-discrete and fully discrete cases.

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arXiv2013

The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation

We consider the cubic nonlinear Schrödinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on $\R^2$ that approximates the dynamics of the frequency modes. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under Fourier transform, as well as several families of explicit solutions. A large part of this work is devoted to a rigorous approximation result that allows to project the long-time dynamics of the limit equation into that of the cubic NLS equation on a box of finite size.

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arXiv2013

Analysis of the Monte-Carlo error in a hybrid semi-lagrangian scheme

We consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. We study the Monte-Carlo error in a simple case, and show that under an anti-CFL condition on the time-step $δt$ and on the mesh size $δx$ and for $N$ - the number of realizations - reasonably large, we control this error by a term of order $\mathcal{O}(\sqrt{δt /N})$. We also provide some numerical experiments to confirm the error estimate, and to expose some examples of equations which can be treated by the numerical method.

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arXiv2012

Analysis of exponential splitting methods for inhomogeneous parabolic equations

We analyze the convergence of the exponential Lie and exponential Strang splitting applied to inhomogeneous second-order parabolic equations with Dirichlet boundary conditions. A recent result on the smoothing properties of these methods allows us to prove sharp convergence results in the case of homogeneous Dirichlet boundary conditions. When no source term is present and natural regularity assumptions are imposed on the initial value, we show full-order convergence of both methods. For inhomogeneous equations, we prove full-order convergence for the exponential Lie splitting, whereas order reduction to 1.25 for the exponential Strang splitting. Furthermore, we give sufficient conditions on the inhomogeneity for full-order convergence of both methods. Moreover our theoretical convergence results explain the severe order reduction to 0.25 of splitting methods applied to problems involving inhomogeneous Dirichlet boundary conditions. We include numerical experiments to underline the sharpness of our theoretical convergence results.

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arXiv2012

Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime

We consider the Klein-Gordon equation in the non-relativistic limit regime, i.e. the speed of light c tending to infinity. We construct an asymptotic expansion for the solution with respect to the small parameter depending on the inverse of the square of the speed of light. As the first terms of this asymptotic can easily be simulated our approach allows us to construct numerical algorithms that are robust with respect to the large parameter c producing high oscillations in the exact solution.

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