Section: New Results

Resolution of linearised systems

Participants : Hubert Alcin, Olivier Allain [Lemma] , Anca Belme, Marianna Braza [IMF-Toulouse] , Alexandre Carabias, Alain Dervieux, Bruno Koobus [Université Montpellier 2] , Carine Moussaed [Université Montpellier 2] , Hilde Ouvrard [IMF-Toulouse] , Stephen Wornom [Lemma] .

The interaction between the sophisticated solution algorithm inside a program and the Automatic Differentiation of the program is a non-trivial issue. An iterative algorithm generally does not store the successive updates of the iterated solution vector. Furthermore, a modern iterative solution algorithm involves several nonlinear processes, like in:

• the evaluation of an optimal step, which results at least from a homographic function of the unknown,

• the orthonormalisation of the updates (Gram-Schmidt method, Hessenberg method).

Applying reverse AD to the iterative solution algorithm produces a linearised iterative algorithm which is transposed and therefore follows a reverse order, with exactly the same number of iterations, and needing exactly each of the iterated state solution vectors. This effect is considerably amplified in the case of the numerical simulation of unsteady phenomena with implicit numerical schemes. For example, the simulation of high Reynolds turbulent flows by a Large Eddy Simulation (LES) requires hundreds of thousands time steps, each of them involving a modern iterative solution algorithm.

In the 4-year ECINADS ANR project, we design more efficient solution algorithms and we examine the questions risen by their reverse differentiation. The application domain is the computation of high Reynolds turbulent flows with LES and hybrid RANS-LES models. The efficiency will be evaluated through the practical scalability on a large number of processors. This efficiency criterion also extends to the scalability of the reverse/adjoint algorithm. ECINADS also addresses the scalable solution of new approximations. ECINADS associates the university of Montpellier 2, the Institut de Mécanique des fluides de Toulouse and Lemma company. The kick off meeting of ECINADS was held at end of 2009.

In 2011, Hubert Alcin has performed a study of deflation and balancing coarse grid methods for a set of scalar models. The methods has been extended to the incompressible Navier-Stokes model in Lemma's software ANANAS by Olivier Allain and to compressible Navier-Stokes by Bruno Koobus and Hubert Alcin. A collection of benchmark tests on these models has been performed and show a good scalability for the tested algorithms. An article is prepared on these results. Hubert Alcin has also studied a novel method for three-level preconditioning. The new method will be extended to compressible flows in cooperation with the Montpellier team (Carine Moussaed and Bruno Koobus).