Laurent Gosse

Maxwellian Decay for Well-balanced Approximations of a Super-characteristic Chemotaxis Model

We focus on the numerical simulation of a one-dimensional model of chemotaxis dynamics (proposed by Greenberg and Alt [Trans. Amer. Math. Soc., 300 (1987), pp. 235-258] in a bounded domain by means of a previously introduced well-balanced (WB) and asymptotic-preserving (AP) scheme [L. Gosse, J. Math. Anal. Appl., (2011)]. We are especially interested in studying the decay onto numerical steady-states for two reasons: (1) conventional upwind schemes have been shown to stabilize onto spurious non-Maxwellian states (with a very big mass flux; see, e.g., [F. Guarguaglini et al., Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), pp. 39-76]) and (2) the initial data lead to a dynamic which is mostly super-characteristic in the sense of [S. Jin and M. Katsoulakis, SIAM J. Appl. Math., 61 (2000), pp. 273-292]; thus the stability results of Gosse do not apply. A reflecting boundary condition which is compatible with the well-balanced character is presented; a mass-preservation property is proved and some results on super-characteristic relaxation are recalled. Numerical experiments with coarse computational grids are presented in detail: they illustrate the bifurcation diagrams of Guarguaglini et al., which relate the total initial mass of cells with the time-asymptotic values of the chemoattractant substance on each side of the domain. It is shown that the WB scheme stabilizes correctly onto zero-mass flow rate (hence Maxwellian) steady-states which agree with the aforementioned bifurcation diagrams. The evolution in time of residues is commented for every considered test case.

A well-balanced scheme for kinetic models of chemotaxis derived from one-dimensional local forward–backward problems

Numerical approximation of one-dimensional kinetic models for directed motion of bacterial populations in response to a chemical gradient, usually called chemotaxis, is considered in the framework of well-balanced (WB) schemes. The validity of one-dimensional models have been shown to be relevant for the simulation of more general situations with symmetry in all but one direction along which appears the chemical attractant gradient. Two main categories are considered depending on whether or not the kinetic equation with specular boundary conditions admits non-constant macroscopic densities for large times. The WB schemes are endowed with the property of having zero artificial viscosity at steady-state; in particular they furnish numerical solutions for which the macroscopic flux vanishes, a feature that more conventional discretizations can miss. A class of equations which admit constant asymptotic states can be treated by a slight variation of the method of Cases elementary solutions originally developed for radiative transfer problems. More involved models which can display concentrations are handled through a different, but closely related, treatment of the tumbling term at the computational grids interfaces. Both types of WB algorithms can be implemented efficiently relying on the Sherman-Morrison formula for computing interface values. Transient and stationary numerical results are displayed for several test-cases.

Error Estimates for Well-Balanced Schemes on Simple Balance Laws

1 Introduction.- 2 Local and global error estimates.- 3 Position-dependent scalar balance laws.- 4 Lyapunov functional for inertial approximations.- 5 Entropy dissipation and comparison with Lyapunov estimates.- 6 Conclusion and outlook.

Resolution of the finite Markov moment problem

Abstract We expose in full detail a constructive procedure to invert the so-called ‘finite Markov moment problem’. The proofs rely on the general theory of Toeplitz matrices together with the classical Newtons relations. To cite this article: L. Gosse, O. Runborg, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Existence, uniqueness and a constructive solution algorithm for a class of finite Markov moment problems

We consider a class of finite Markov moment problems with an arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the nonunique solution families. Moreover, we present a constructive algorithm to solve the moment problems numerically and prove that the algorithm computes the right solution.

A well-balanced scheme able to cope with hydrodynamic limits for linear kinetic models

Abstract 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.

A Donoho–Stark criterion for stable signal recovery in discrete wavelet subspaces

Abstract We derive a sufficient condition by means of which one can recover a scale-limited signal from the knowledge of a truncated version of it in a stable manner following the canvas introduced by Donoho and Stark (1989) [4] . The proof follows from simple computations involving the Zak transform, well-known in solid-state physics. Geometric harmonics (in the terminology of Coifman and Lafon (2006) [22] ) for scale-limited subspaces of L 2 ( R ) are also displayed for several test-cases. Finally, some algorithms are studied for the treatment of zero-angle problems.

Viscous Equations Treated with \(\mathcal{L}\)-Splines and Steklov-Poincaré Operator in Two Dimensions

Well-balanced schemes, nowadays well-known for 1D hyperbolic equations with source terms and systems of balance laws, are extended to strictly parabolic equations, first in 1D, then in 2D on Cartesian computational grids. The construction heavily relies on a particular type of piecewise-smooth interpolation of discrete data known as \(\mathcal{L}\)-splines. In 1D, several types of widely-used discretizations are recovered as particular cases, like the El-Mistikawy-Werle scheme or Scharfetter-Gummel’s. Moreover, a distinctive feature of our 2D scheme is that dimensional-splitting never occurs within its derivation, so that all the multi-dimensional interactions are kept at the discrete level. This leads to improved accuracy, as illustrated on several types of drift-diffusion equations.

A Semiclassical Coupled Model for the Transient Simulation of Semiconductor Devices

We consider the approximation of a microelectronic device corresponding to an

Filtered Gradient Algorithms for Inverse Design Problems of One-Dimensional Burgers Equation

n^+-n-n^+