Langevin dynamic for the 2D Yang–Mills measure

Publications mathématiques de l'IHÉS - It is well known that a real analytic symplectic diffeomorphism of the $2d$ -dimensional disk ( $d\geq 1$ ) admitting the origin as a...     

One-step and extrapolation methods for differential-algebraic systems

Summary The paper analyzes one-step methods for differential-algebraic equations (DAE) in terms of convergence order. In view of extrapolation methods, certain perturbed asymptotic expansions are shown to hold. For the special DAE extrapolation solver based on the semi-implicit Euler discretization, the perturbed order pattern of the extrapolation tableau is derived in detail. The theoretical results lead to modifications of the known code. The efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion.     

Spectral Semi-discretisations of Weakly Non-linear Wave Equations over Long Times

The long-time behaviour of spectral semi-discretisations of weakly non-linear wave equations is analysed. It is shown that the harmonic actions are approximately conserved for the semi-discretised system as well. This permits to prove that the energy of the wave equation along the interpolated semi-discrete solution remains well conserved over long times and close to the Hamiltonian of the semi-discrete equation. Although the momentum is no longer an exact invariant of the semi-discretisation, it is shown to be approximately conserved. All these results are obtained with the technique of modulated Fourier expansions. Dedicated to Professor Arieh Iserles on the Occasion of his Sixtieth Birthday.     

Homogenization of periodic linear degenerate PDEs

It is well known under the name of ‘periodic homogenization’ that, under a centering condition of the drift, a periodic diffusion process on Rd converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set. As a consequence, it is no longer the case that the effective diffusivity matrix is necessarily non-degenerate. It turns out that, provided that some very weak regularity conditions are met, the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion. In particular, this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank. We also discuss the application of our results to the homogenization of a class of elliptic and parabolic PDEs.     

Achieving Brouwer’s law with implicit Runge–Kutta methods

In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems. For implicit Runge–Kutta methods, a standard implementation shows an unexpected propagation. We propose a modification that reduces the effect of round-off and shows a qualitative and quantitative improvement for an accurate integration over long times. AMS subject classification (2000)  65L06, 65G50, 65P10     

A version of Hörmander’s theorem for the fractional Brownian motion

It is shown that the law of an SDE driven by fractional Brownian motion with Hurst parameter greater than 1/2 has a smooth density with respect to Lebesgue measure, provided that the driving vector fields satisfy Hörmander’s condition. The main new ingredient of the proof is an extension of Norris’ lemma to this situation.     

Equilibria of Runge-Kutta methods

Summary It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.     

Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations

The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the St?rmer–Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included. AMS subject classification (2000) 65L05, 65P10     

Modulated Fourier Expansions of Highly Oscillatory Differential Equations

Modulated Fourier expansions are developed as a tool for gaining insight into the long-time behavior of Hamiltonian systems with highly oscillatory solutions. Particle systems of Fermi–Pasta–Ulam type with light and heavy masses are considered as an example. It is shown that the harmonic energy of the highly oscillatory part is nearly conserved over times that are exponentially long in the high frequency. Unlike previous approaches to such problems, the technique used here does not employ nonlinear coordinate transforms and can therefore be extended to the analysis of numerical discretizations.