Laurent Gosse - Academia.edu (original) (raw)

Papers by Laurent Gosse

ANNALI DELL'UNIVERSITA' DI FERRARA

By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as s... more By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linear wave equations. Accordingly, the Kirchhoff expression of the exact solutions is recast as a timemarching, Lax-Wendroff type, numerical scheme for which consistency with one-dimensional upwinding is checked. This discretization, involving spherical means, is set up on a 2D uniform Cartesian grid, so that the resulting numerical fluxes can be shown to be conservative. Moreover, semi-discrete stability in the H s norms and vorticity dissipation are established, along with practical second-order accuracy. Finally, some relations with former "shape functions" and "symmetric potential schemes" are highlighted.

This paper investigates the behavior of numerical schemes for nonlinear conservation laws with so... more This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided.

J Comput Phys, 2006

This work is concerned with the semiclassical approximation of the Schrö dinger-Poisson equation ... more This work is concerned with the semiclassical approximation of the Schrö dinger-Poisson equation modeling ballistic transport in a 1D periodic potential by means of WKB techniques. It is derived by considering the mean-field limit of a N-body quantum problem, then K-multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for PauliÕs exclusion principle. Numerical experiments display the behaviour of self-consistent wave packets and screening effects.

Numerische Mathematik, Jan 31, 2002

This paper is devoted to both theoretical and numerical study of a system involving an eikonal eq... more This paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of Hamilton-Jacobi type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical limit for the Schrödinger equation. We first state an existence and uniqueness result in the framework of viscosity and duality solutions. Then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions to ensure convergence. As an illustration, some practical computations are provided.

Computers Mathematics With Applications, Sep 1, 2010

We consider a rather simple algorithm to address the fascinating field of numerical extrapolation... more We consider a rather simple algorithm to address the fascinating field of numerical extrapolation of (analytic) band-limited functions. It relies on two main elements: namely, the lower frequencies are treated by projecting the known part of the signal to be extended onto the space generated by "Prolate Spheroidal Wave Functions" (PSWF, as originally proposed by Slepian), whereas the higher ones can be handled by the recent so-called "Compressive Sampling" (CS, proposed by Candès) algorithms which are independent of the largeness of the bandwidth. Slepian functions are recalled and their numerical computation is explained in full detail whereas Compressive Sampling techniques are summarized together with a recent iterative algorithm which has been proved to work efficiently on so-called "compressible signals" which appear to match rather well the class of smooth bandlimited functions. Numerical results are displayed for both numerical techniques and the accuracy of the process consisting in putting them altogether is studied for several test-signals.

We are concerned with efficient numerical simulation of the radiative transfer equations. To this... more We are concerned with efficient numerical simulation of the radiative transfer equations. To this end, we follow the Well-Balanced approach's canvas and reformulate the relaxation term as a nonconservative product regularized by steady-state curves while keeping the velocity variable continuous. These steady-state equations are of Fredholm type. The resulting upwind schemes are proved to be stable under a reasonable parabolic CFL condition of the type ∆t ≤ O(∆x 2) among other desirable properties. Some numerical results demonstrate the realizability and the efficiency of this process.

ABSTRACT The aim of this note is to derive an a-priori error estimate for the approximate solutio... more ABSTRACT The aim of this note is to derive an a-priori error estimate for the approximate solution generated by means of the numerical scheme proposed in [5] and extensively studied in [4]. This scheme belongs to the class of well-balanced schemes introduced by Greenberg and LeRoux [6]; its main features are its robustness and its ability to preserve the correct steady-states of the continuous problem. Here, we show that it is endowed with the usual O( p Deltax) error in most cases. Estimation d'erreur a priori pour un sch'ema 'equilibre adapt'e aux lois de conservation scalaires non-homog`enes R'esum'e Le but de cette note est d'obtenir une estimation d'erreur a-priori pour les solutions approch'ees g'en'er'ees grace au sch'ema num'erique propos'e dans [5] et 'etudi'e dans [4]. Ce sch'ema appartient `a la classe des sch'emas 'equilibre introduits par Greenberg et LeRoux [6]; ses principales caract'eristiques sont sa robustesse et sa capacit'e `a pr'eserver les 'etats stationnaires correct...

ABSTRACT We prove Oleĭnik-type decay estimates for entropy solutions of n×n strictly hyperbolic s... more ABSTRACT We prove Oleĭnik-type decay estimates for entropy solutions of n×n strictly hyperbolic systems of balance laws built out of a wave-front tracking procedure inside which the source term is treated as a nonconservative product localized on a discrete lattice. 1.

A posteriori L 1 error estimates (in the sense of [15, 26]) are derived for both well-balanced (W... more A posteriori L 1 error estimates (in the sense of [15, 26]) are derived for both well-balanced (WB) and fractional-step (FS) numerical approximations of the unique weak solution of the Cauchy problem for the 1D semilinear damped wave equation. For setting up the WB algorithm, we proceed by rewriting it under the form of an elementary 3 × 3 system which linear convective structure allows to reduce the Godunov scheme with optimal Courant number (corresponding to ∆t = ∆x) to a wavefront-tracking algorithm free from any step of projection onto piecewise constant functions. A fundamental difference in the total variation estimates is proved, which partly explains the discrepancy of the FS method when the dissipative (sink) term displays an explicit dependence in the space variable. Numerical tests are performed by means of stationary exact solutions of the linear damped wave equation.

SpringerBriefs in Mathematics, 2015

Kinetic and Related Models, 2012

In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for whic... more In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear flow effects follow a scalar integro-differential equation whereas the heat transfer is described by a 2 × 2 coupled system. This simplification allows to set up the well-balanced method, involving non-conservative products regularized by solutions of the stationary equations, in order to produce numerical schemes which do stabilize in large times and deliver accurate approximations at numerical steady-state. Boundary-value problems for the stationary equations are solved by the technique of "elementary solutions" at the continuous level and by means of the "analytical discrete ordinates" method at the numerical one. Practically, a comparison with a standard time-splitting method is displayed for a Couette flow by inspecting the shear stress which must be a constant at steady-state. Other test-cases are treated, like heat transfer between two unequally heated walls and also the propagation of a sound disturbance in a gas at rest. Other numerical experiments deal with the behavior of these kinetic models when the Knudsen number becomes small. In particular, a test-case involving a computational domain containing both rarefied and fluid regions characterized by mean free paths of different magnitudes is presented: stabilization onto a physically correct steady-state free from spurious oscillations is observed.

SIAM Journal on Applied Mathematics, 2015

An elementary model of 1 + 1-dimensional general relativity, known as "R = T " and mainly develop... more An elementary model of 1 + 1-dimensional general relativity, known as "R = T " and mainly developed by Mann et al. [44, 45, 48, 50, 51, 56, 63], is set up in various contexts. Its formulation, mostly in isothermal coordinates, is derived and a relativistic Euler system of self-gravitating gas coupled to a Liouville equation for the metric's conformal factor is deduced. First, external field approximations are carried out: both a Klein-Gordon equation is studied along with its corresponding density, and a Dirac one inside an hydrostatic gravitational field induced by a static, piecewise constant mass repartition. Finally, the coupled Euler-Liouville system is simulated, by means of a locally inertial Godunov scheme: the gravitational collapse of a static random initial distribution of density is displayed. Well-balanced discretizations rely on the treatment of source terms at each interface of the computational grid, hence the metric remains flat in every computational cell.

ANNALI DELL'UNIVERSITA' DI FERRARA, 2012

Efficient recovery of smooth functions which are s-sparse with respect to the base of so-called P... more Efficient recovery of smooth functions which are s-sparse with respect to the base of so-called Prolate Spheroidal Wave Functions from a small number of random sampling points is considered. The main ingredient in the design of both the algorithms we propose here consists in establishing a uniform L ∞ bound on the measurement ensembles which constitute the columns of the sensing matrix. Such a bound provides us with the Restricted Isometry Property for this rectangular random matrix, which leads to either the exact recovery property or the "best s-term approximation" of the original signal by means of the ℓ 1 minimization program. The first algorithm considers only a restricted number of columns for which the L ∞ holds as a consequence of the fact that eigenvalues of the Bergman's restriction operator are close to 1 whereas the second one allows for a wider system of PSWF by taking advantage of a preconditioning technique. Numerical examples are spread throughout the text to illustrate the results.

Hyperbolic Problems: Theory, Numerics, Applications, 2001

Applied Mathematics Letters, 2015

Well-balanced schemes were introduced to numerically enforce consistency with long-time behavior ... more Well-balanced schemes were introduced to numerically enforce consistency with long-time behavior of the underlying continuous PDE. When applied to linear kinetic models, like the Goldstein-Taylor system, this construction generates discretizations which are inconsistent with the hydrodynamic stiff limit (despite it captures diffusive limits quite well). A numerical hybridization, taking advantage of both time-splitting (TS) and well-balanced (WB) approaches is proposed in order to fix this defect: numerical results show that resulting composite schemes improve rendering of macroscopic fluxes while keeping a correct hydrodynamic stiff limit.

Multiscale Modeling & Simulation, 2014

When numerically simulating a kinetic model of n + nn + semiconductor device, obtaining a constan... more When numerically simulating a kinetic model of n + nn + semiconductor device, obtaining a constant macroscopic current at steady-state is still a challenging task. Part of the difficulty comes from the multiscale, discontinuous nature of both p|n junctions which create spikes of electric field and enclose a channel where corresponding depletion layers glue together. The kinetic formalism furnishes a model holding inside the whole domain, but at the price of strongly-varying parameters. By concentrating both the electric acceleration and the linear collision terms at each interface of a Cartesian computational grid, we can treat them by means of a Godunov scheme involving 2 types of scattering matrices. Combining both these mechanisms into a global S-matrix can be achieved thanks to "Redheffer's star-product". Assuming that the resulting S-matrix is stochastic permits to prove maximum principles under a mild CFL restriction. Numerical illustrations of collisional Landau damping and various n + nn + devices are provided on coarse grids.