Section: New Results

Numerical methods

Inspired by the quantitative analysis of [15] and [17] , Z. Habibi (former SIMPAF post-doctoral fellow) and A. Gloria introduced in [13] a general method to reduce the so-called resonance error in numerical homogenization, both at the levels of the approximation of the homogenized coefficients and of the correctors. This method significantly extends [2] . The method relies on the introduction of a massive term in the corrector equation and of a systematic use of Richardson extrapolation. In the three academic examples of heterogeneous coefficients (periodic, quasiperiodic, and Poisson random inclusions), the method yields optimal theoretical and empirical convergence rates, and outperforms most of the other existing methods.

In [11] , G. Dujardin and P. Lafitte (ECP) published a result on the asymptotic behavior of splitting schemes applied to multiscale systems which have strongly attracting equilibrium states. They proposed a definition of the asymptotic order of such schemes and proved on examples of ODEs and PDEs systems that one can achieve high asymptotic order with such schemes, provided sufficient conditions are fulfilled.

In [25] , G. Dujardin proposed to use high order methods for the numerical simulation of rotating Bose-Einstein condensates. With his co-authors, he developed exponential Runge-Kutta methods and Lawson method for this problem and he analyzed the convergence order of these methods. In particular, they proved that one can achieve maximal order $2s$ with methods with $s$-stages. They also supported their analysis with numerical experiments carried out in physically realistic simulations.