Long-time behavior of Ginzburg-Landau systems far from equilibrium

Using singular-perturbation techniques, we study the stability of modulated structures generated by driving Ginzburg-Landau systems far from equilibrium. We show that, far from equilibrium, the steady-state behavior is controlled by an effective Lagrangian which possesses the same functional form as the original free energy but with renormalized coefficients. We study both linear and nonlinear sources and determine their influence on the long-term stability of the bifurcating solutions

Advanced stability theory analyses for laminar flow control

Recent developments of the SALLY computer code for stability analysis of laminar flow control wings are summarized. Extensions of SALLY to study three dimensional compressible flows, nonparallel and nonlinear effects are discussed

Spectral Methods for Numerical Relativity

Version published online by Living Reviews in Relativity.International audienceEquations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers

Navier-Stokes solver using Green's functions I: channel flow and plane Couette flow

Numerical solvers of the incompressible Navier-Stokes equations have reproduced turbulence phenomena such as the law of the wall, the dependence of turbulence intensities on the Reynolds number, and experimentally observed properties of turbulence energy production. In this article, we begin a sequence of investigations whose eventual aim is to derive and implement numerical solvers that can reach higher Reynolds numbers than is currently possible. Every time step of a Navier-Stokes solver in effect solves a linear boundary value problem. The use of Green's functions leads to numerical solvers which are highly accurate in resolving the boundary layer, which is a source of delicate but exceedingly important physical effects at high Reynolds numbers. The use of Green's functions brings with it a need for careful quadrature rules and a reconsideration of time steppers. We derive and implement Green's function based solvers for the channel flow and plane Couette flow geometries. The solvers are validated by reproducing turbulent signals which are in good qualitative and quantitative agreement with experiment