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Section: New Results

Resolution of linearised systems and efficiency

Participants : Olivier Allain [Lemma] , Gautier Brèthes, Alain Dervieux, Bruno Koobus [Université Montpellier 2] , Emmanuelle Itam [Université Montpellier 2] , Vincent Levasseur [Lemma] , Stephen Wornom [Lemma] .

For Fluid Mechanics as well as for Structural Mechanics, an implicit time-advancing is mandatory. It can be applied efficiently if the large systems involved are solved with a good parallel algorithm. In the 90's, a generation of solution algorithms was devised on the basis of Domain Decomposition Methods (DDM). For complex models (compressible flows...), Schwarz DDM were combined with quasi-Newton algorithms such as GMRES. These are for example Restrictive Additive Schwarz (RAS), which is used in our platform AIRONUM. RAS was developed by Cai, Farhat and others. RAS is an ancestor of the widely used class of Newton-Krylov-Schwarz (NKS) algorithms. For hundreds of processors many versions of NKS, and in particular RAS, are almost scalable (convergence rate independant of the number of processors). But scalability vanishes for a medium-large number of processors (thousands). In the ANR ECINADS, coordinated by Ecuador, a Coarse-Grid Deflated RAS was developped: iteration-wise scalability holds for all parts, except for the coarse grid direct solver, which concerns a much smaller problem. Effective Convergence Scalability (ECS) was confirmed up to 2048 processors. Beyond this level the asymptotic complexity of the coarse-grid direct solver becomes predominant and ECS is lost. In other words, with a Coarse-Grid Deflated RAS, the size of the coarse grid problem which is solved by a direct algebraic solver must be limited in order to enjoy ECS. For finer meshes, the coarse system cannot be finer, and the efficiency is lower. It is then natural to consider intermediate meshes on which iterative solvers will be applied. In the ANR MAIDESC, Gautier Brèthes has defined a multi-mesh Full MultiGrid (FMG) algorithm adapted to anisotropic mesh adaptation. In 2015, the method has been extended to MPI-based massive parallelism, in cooperation with the Lemma team for the computation of incompressible flows. As a perspective, our parallel MG can be complemented with the previous version of the solver (deflated RAS) for a higher degree of scalability.

A second issue which we addressed is the use of explicit time advancing. Many unsteady flows have to be computed with explicit time advancing. A single explicit time step is of a low cost and can be highly accurate. Explicit time advancing is mandatory for wave propagation: blast shocks of vortices in wakes. However the meshes used may involve small regions in which the explicit time step should be very small and large regions in which such a small time step is a waste. The family of time-advancing methods in which unsteady phenomena are computed using different time steps in different regions is called the multirate methods. In our cooperation with University of Montpellier, a novel multirate method using cell-agglomeration has been designed and developed in our AIRONUM platform. An article is in preparation. This work takes place in the ANR MAIDESC programme.