Section: New Results

Resolution of linearised systems

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

Increased sophistication of solution algorithms pose a challenge to Automatic Differentiation. Time-stepping iterations create numerous updates of the iterated solution vector. Other additional nonlinear iterative processes occur such as:

• 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).

Adjoint-mode AD applied to these algorithms produces a “linearised iterative algorithm” which is transposed and therefore follows the original iterations in the reverse order, needing each of the iterated state solution vectors. One such extreme case is the simulation of unsteady phenomena with implicit numerical schemes: simulating high Reynolds turbulent flows by a Large Eddy Simulation (LES) and RANS-LES models requires hundreds of thousands time steps, each of them involving a modern iterative solution algorithm. This is the case targetted by the 4-year ANR project “ECINADS”, jointly with university of Montpellier 2, the Institut de Mécanique des fluides de Toulouse and Lemma company, started in 2009.

In ECINADS, we design more efficient solution algorithms and we examine the questions risen by their adjoint differentiation. Our goal is practical scalability of the direct simulation and of its adjoint on a large number of processors. ECINADS also addresses the scalable solution of new approximations.

In 2012, the novel three-level method studied by H. Alcin on a model problem has been extended to compressible viscous flows by B. Koobus and C. Moussaed from university of Montpellier.

Hubert Alcin, Bruno Koobus, Olivier Allain and Alain Dervieux published their work on a two-level Schwarz algorithm in IJNMFD [12] . H. Alcin has presented his work in the Parallel CFD conference of Altlanta [14] . H. Alcin wrote his thesis [11] , defended in december, on the three main subjects of ECINADS: the two- and three-level Schwarz algorithms, Automatic Differentiation and mesh adaptation.