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This paper introduces the F3ORNITS non-iterative co-simulation algorithm in which F3 stands for the 3 flexible aspects of the method: flexible polynomial order representation of coupling variables, flexible time-stepper applying variable co-simulation step size rules on subsystems allowing it and flexible scheduler orchestrating the meeting times among the subsystems and capable of asynchronousness when subsystems constraints requires it. The motivation of the F3ORNITS method is to accept any kind of co-simulation model, including any kind of subsystem, regardless on their available capabilities. Indeed, one the major problems in industry is that the subsystems usually have constraints or lack of advanced capabilities making it impossible to implement most of the advanced co-simulation algorithms on them. The method makes it possible to preserve the dynamics of the coupling constraints when necessary as well as to avoid breaking C1 smoothness at communication times, and also to adapt the co-simulation step size in a way that is robust both to zero-crossing variables (contrary to classical relative error-based criteria) and to jumps. Two test cases are presented to illustrate the robustness of the F3ORNITS method as well as its higher accuracy than the non-iterative Jacobi coupling algorithm (the most commonly used method in industry) for a smaller number of co-simulation steps.

IFOSMONDI iterative algorithm for implicit co-simulation of coupled physical systems (introduced by the authors in july 2019 during the Simultech conference, p.176-186) enables us to solve the nonlinear coupling function while keeping the smoothness of interfaces without introducing a delay. Moreover, it automatically adapts the size of the steps between data exchanges among the systems according to the difficulty of the solving of the coupling constraint. The latter was solved by a fixed-point algorithm in the original implementation whereas this paper introduces the JFM version (standing for Jacobian-Free Methods). Most implementations of Newton-like methods require a jacobian matrix which can be difficult to compute in the co-simulation context, except in the case where the interfaces are represented by a Zero-Order-Hold (ZOH). As far as IFOSMONDI coupling algorithm uses Hermite interpolation for smoothness enhancement (up to Third-Order-Hold), we propose hereafter a new formulation of the non-linear coupling function including both the values and the time-derivatives of the coupling variables. This formulation is well designed for solving the coupling through jacobian-free Newton type methods. Consequently, successive function evaluations consist in multiple simulations of the systems on a co-simulation time step using rollback. The orchestrator-workers structure of the algorithm enables us to combine the PETSc framework on the orchestrator side for the non-linear Newton-type solvers with the parallel integrations of the systems on the workers side thanks to MPI processes. Different nonlinear methods will be compared to one another and to the original fixed-point implementation on a newly proposed 2-systems academic test-case (mass-spring-damper type) with direct feedthrough on both sides.

In a co-simulation context, interconnected systems of differential equations are solved separately but they regularly communicate data to one another during these resolutions. Iterative co-simulation methods have been developed in order to enhance both stability and accuracy. Such methods imply that the systems must integrate one or more times per co-simulation step (the interval between two consecutive communications) in order to find the best satisfying interface values for exchanged data (according to a given coupling constraint). This requires that every system involved in the modular model is capable of rollback: the ability to re-integrate a time interval that has already been integrated with different input commands. In a paper previously introduced by Eguillon et al. in 2022, the COSTARICA process is presented and consists in replacing the non-rollback-capable systems by an estimator on the non-last integrations of the iterative process. The MISSILES algorithm, introduced in this paper, consists in applying the COSTARICA process on every system of a modular model simulated with the IFOSMONDI-JFM iterative co-simulation method (introduced by Eguillon et al. in 2021). Indeed, in this case, the iterative part on the estimators of each system can be avoided as the global resolution on a co-simulation step can be written as a single global linear system to solve. Consequently, MISSILES is a non-iterative method that leads to the same solution than the IFOSMONDI-JFM iterative co-simulation method applied to systems using the COSTARICA process to emulate the rollback.

With the introduction of renewable energy sources into the power system, transmission system operators need extremely reliable and flexible simulation tools that should also be fast. Sinusoidal predictor method, aims to meet this need by simulating the full-waveform of AC power systems by integrating the arising systems of DAE by trying to free the time step limitation introduced by the frequency of some electrical components. The searched solution is split in a predicted sinusoidal part and a corrector DAE solution part. This paper presents the mathematical convergence of this approach leading to the choice of computing Fourier coefficients by minimizing a function that captures the distance of the desired sinusoidal part of the solution with respect to its oscillating steady state. Using the Dynaωo framework that aims to completely separate the modeling and the solving aspects of power systems, It details latest developments of an add-on in the IDA solver of SUNDIALS and the implementation optimisations. Results on performance speed up of the method computational time using much larger time steps than those used by classical DAE integration methods.

International audience

This paper deals with the representation of the trace of iterative Schwarz solutions at the interfaces of domain decomposition to approximate adaptively the interface error operator. This allows to build a cost-effectively accelerating of the convergence of the iterative method by extending to the vectorial case the Aitken’s accelerating convergence technique. The first representation is based on the building of a nonuniform discrete Fourier transform defined on a non-regular grid. We show how to construct a Fourier basis of dimension N+1 on this grid by building numerically a sesquilinear form, its exact accuracy to represent trigonometric polynomials of degree N / 2, and its spectral approximation property that depends on the continuity of the function to approximate. The decay of Fourier-like modes of the approximation of the trace of the iterative solution at the interfaces provides an estimate to adaptively select the modes involved in the acceleration. The drawback of this approach is to be dependent on the continuity of the trace of the iterated solution at the interfaces. The second representation, purely algebraic, uses a singular value decomposition of the trace of the iterative solution at the interfaces to provide a set of orthogonal singular vectors of which the associated singular values provide an estimate to adapt the acceleration. The resulting Aitken-Schwarz methodology is then applied to large scale computing on 3D linear Darcy flow where the permeability follows a log normal random distribution. Cet acte traite de la représentation des solutions itérées aux interfaces de la méthode de décomposition de domaine de type Schwarz afin d’approximer de manière adaptative son opérateur d’erreur aux interfaces des sous domaines. Ceci permet de construire de manière économique l’accélération de la convergence de la méthode itérative en étendant la technique d’accélération de la convergence de Aitken au cas vectoriel. La première représentation est fondée sur la construction d’une transformée de Fourier discrète non uniforme définie sur un maillage non régulier. Nous montrons comment construire une base de Fourier de dimension N+1 sur ce maillage à partir de la construction numérique d’une forme sesquilinéaire, son exactitude pour les polynômes trigonométriques de degré N/2, et numériquement sa capacité d’approximation spectrale dépendante de la continuité de la fonction à approximer. La décroissance des modes de Fourier de la trace de la solution itérée aux interfaces nous fournis une estimation pour sélectionner de manière adaptive les modes intervenant dans l’accélération. Le défaut de cette approche est d’être dépendant de la continuité de la solution itérée aux interfaces. La deuxième représentation, purement algébrique, utilise une décomposition en valeurs singulières des solutions itérées aux interfaces pour fournir un ensemble de vecteurs singuliers orthogonaux dont les valeurs singulières associées fournissent une estimation pour adapter l’accélération. La méthode Aitken-Schwarz résultante est ensuite appliquée au calcul à grande échelle pour résoudre un écoulement 3D en milieu poreux modélisé par l’équation de Darcy linéaire où la perméabilité suit une distribution aléatoire log-normale.

International audience; This paper is devoted to the acceleration by Aitken's technique of the convergence of the Schwarz domain decomposition method applied to large scale 3D problems with non-separable linear operators. These operators come from the discretization of groundwater flow problems modeled by the linear Darcy equation, where the permeability field is highly heterogeneous and randomly generated. To be computationally efficient, a low-rank approximation of the Aitken's formula is computed from the singular value decomposition of successive iterated solutions on subdomains interfaces. Numerical results explore the efficiency of the solver with respect to the random distribution parameters, and specific implementations of the acceleration are compared for large scale 3D problems. These results confirm the numerical behavior of the methodology obtained on 2D Darcy problems (Tromeur-Dervout D. Meshfree adaptive Aitken-Schwarz domain decomposition with application to Darcy flow. Comput Sci Eng Technol 2009;21:217-50).

International audience; We developed a parallel time domain decomposition method to solve systems of ODEs based on the Aitken-Schwarz domain decomposition method only in time (Linel P, Tromeur-Dervout D. Aitken-Schwarz and schur complement methods for time domain decomposition. In: Parallel computing: from multicores and GPU's to Petascale. Advances in parallel computing, vol. 19; 2010. p. 75-82). The method transforms the initial time value problem into a time boundary values problem. This paper details the proof of the pure linear convergence of this DDM in case of linear system of ODEs and provides an optimized MPI parallel implementation when a regular size splitting of the time interval is performed. Then an extension of the method to the linear heat partial derivative equation is provided with a special care on boundary conditions to impose. A Fourier convergence analysis of this Schwarz DDM in time for this problem allows to limit the acceleration by Aitken's technique to Fourier low modes only.

This paper details the software implementation of the ARAS preconditioning technique (Dufaud T, Tromeur-Dervout D. Aitken's acceleration of the Restricted Additive Schwarz preconditioning using coarse approximations on the interface. CR Math Acad Sci Paris 2010;348(13-14):821-4), in the PETSc framework. Especially, the PETSc implementation of interface operators involved in ARAS and the introduction of a two level of parallelism in PETSc for the RAS are described. The numerical and parallel implementation performances are studied on academic and industrial problems, and compared with the RAS preconditioning. For saving computational time on industrial problems, the Aitken's acceleration operator is approximated from the singular values decomposition technique of the RAS iterate solutions.

International audience; This paper details the software implementation of the ARAS preconditioning technique (Dufaud T, Tromeur- Dervout D. Aitken's acceleration of the Resctricted Additive Schwarz preconditioning using coarse approximations on the interface. CR Math Acad Sci Paris 2010;348(13-14):821-4), in the PETSc framework. Especially, the PETSc implementation of interface operators involved in ARAS and the introduction of a two level of parallelism in PETSc for the RAS are described. The numerical and parallel implementation performances are studied on academic and industrial problems, and compared with the RAS preconditioning. For saving computational time on industrial problems, the Aitken's acceleration operator is approximated from the singular values decomposition technique of the RAS iterate solutions.

International audience; This Note is concerned with the severely ill-posed Cauchy-Helmholtz problem. This Cauchy problem being rephrased through an "interfacial" equation, we resort to an Aitken-Schwarz method for solving this equation. Numerical trials highlight the efficiency of the present method.

A method for the acceleration of a fully implicit solution of nonlinear unsteady boundary value problem is presented. The principle of acceleration is for provide to the inexact Newton backtracking method a better initial guess, for the current time step, than the conventional choice from the previous time step. This initial guess is built on the reduced model obtained by a proper orthogonal decomposition of solutions at the previous time steps. This approach is appealing to GRID computing: spare processors may help to improve the numerical efficiency and to manage the computing in a reliable way.

International audience; This paper presents a method that speeds up the solution of unsteady nonlinear problems. When a Newton–Krylov method is used, the most time-consuming part of the numerical simulation is the solution of the successive linear systems. Generally, there are only slight changes between two consecutive Jacobian matrices. Thus, in order to save computations, it is possible to use the same preconditioning matrix for a few successive linear systems, but the convergence of the Krylov method is slowed down. In the case of a preconditioner based on domain decomposition, one can update only the parts of the preconditioner associated to certain subdomains, keeping the other ones constant. In order to achieve an efficient parallel implementation, we propose to add processes dedicated to the asynchronous update of some parts of the preconditioner. Numerical results are provided for a reaction–diffusion problem and for a model problem based on the lid-driven cavity. They show that this addition of processes can speed up the computation and improve the robustness.

Metacomputing is a method of using the GRID, which originated in the US and quickly was also picked up by European and Japanese researchers where a number of challenging projects were aiming at the exploitation of such distributed resources. Especially the GLOBUS project (The globus project: a status report, Proceedings IPPS/SPDP'98 Heterogeneous Computing Workshop, 1998, pp. 4-18) has contributed to the success of metacomputing substantially. However, high latency and low bandwidth have made people doubt the feasibility of this concept for big simulation codes. In this paper we present new numerical methods that help to exploit such configurations and overcome the problems of low network performance. To proof the feasibility of our approach we show results of simulations in an innovative GRID environment of supercomputers.

International audience; This paper is devoted to the solution of nonlinear time-dependant partial differential equations arising in CFD using Broyden's method in the parallel computing framework. We first use Broyden's method in the context of the domain decomposition: we propose to update the Restricted Additive Schwarz preconditioner from one Newton iteration to another when a Newton-Krylov method is used. We also investigate a time-pipelining method where Broyden's method is used as a solver of the nonlinear problem of each time step.

Colloque avec actes et comité de lecture. Internationale.; International audience; Nous présentons une méthode parallèle pour résoudre l'équation de Darcy 3D où le champ de perméabilité varie aléatoirement suivant une distribution log normale et avec de fortes amplitudes dans des domaines discrétisés de 10^8 à 10^9 inconnues. Cette technique de décomposition de domaine de type Aitken-Schwarz conduit à un parallélisme à deux niveaux où les problèmes locaux sont résolus par multigrille algébrique parallèle (AGMG de Y. Notay). L'influence sur la construction de l'accélération d' Aitken de l'espace d'approximation pour représenter la solution sur les interfaces sera discuté.

Summary. The parareal scheme (resp. PITA algorithm) proposed in [3] (resp. [2]) considers two levels of grids in time in order to split the domain in time-subdomains. A prediction of the solution is computed on the fine grid in parallel. Then at each interface between the time subdomains, the solution makes a jump between the previous initial boundary value (IBV) of the next time-subdomain . A correction of the IBV for the next fine grid iteration is then computed on the coarse grid in time. In this paper, we study adaptivity in the time slice decomposition based on an a posteriori numerical estimation obtained from the time step behavior on coarse grids. The outline of this paper is as follows: in section 1, the original parareal method is recalled and it is shown that it is a particular case of the multiple shooting method of Deuflhard [1]. Then in section 2, the definition of the size of the fineness of the grids is slightly modified in order to introduce adaptivity within the parareal algorithm for the time stepping, the number of subdomains, and the time decomposition. This adaptivity leads to an improvement of the method and enables us to solve moderately stiff nonlinear ODEs problems. Nevertheless for very stiff problems as the Oregonator model, it fails even with the introduced adaptivity. This leads to develop in section 3 an adaptive parallel extrapolation method, based on a posteriori numerical assessment, which obtains results for this stiff problem.

For flow problems in multi-layered porous media, one can define a natural non-overlapping domain decomposition (DD). The simplest way to obtain DDMs is to distribute interface conditions (pressure and flux continuity) for each pair of adjacent subdomains and to use the Dirichlet-Neumann (D-N) algorithm. A different way is the use of two Robin conditions (RC) also distributed for each subdomain (Robin-Method). The main inconvenience of both methods is that the convergence is not ensured. To obtain efficient methods, we retain from previous works two basic ideas: an acceleration of Aitken type for the D-N algorithm and finding optimized coefficients for the Robin-Method. In the present paper, we analyze these improved algorithms in 1-D and 2-D framework for flow problems in heterogeneous porous media and we present a numerical comparison.

Modern power systems time-domain simulations require important computational resources due to the resulting differential algebraic systems dimension and stiffness. In addition, some simulated signals oscillation frequency dramatically limits the classical schemes step size, even in steady-state during which they are close to sinusoids oscillating at system nominal frequency. That's why the sinusoidal predictors method proposed in this thesis aims at taking this property into account in order to enhance solver performances while controlling the integration error. It consists in decomposing the solution into two parts: a sinusoid, whose Fourier coefficients are fixed for each time integration interval and then updated by parametric estimation, and a correction term on which the DAE system is rewritten and solved using an adaptive step size integration scheme. A particular focus has been given on the estimator choice, given its precision direct impact on the step size and its indirect effect on the global method stability. The finally developed estimator consists in computing Fourier coefficients minimizing a system stationarity measurement. As it converges in steady-state, the correction term is progressively damped, which enables to considerably increase the step size. This method, integrated into the reference solver SUNDIALS IDA and interfaced with an industrial simulation engine, enables to very significantly accelerate simulations in comparison with a classical implementation, Simuler temporellement les réseaux électriques modernes requiert d'importants moyens de calcul de par la dimension et la raideur des systèmes différentiels algébriques résultants. De plus, la fréquence d'oscillation de certains signaux simulés contraint fortement le pas d'intégration des schémas classiques, y compris en régime établi où ils sont proches de sinusoïdes oscillant à la fréquence nominale du système. L'objectif de la méthode des prédicteurs sinusoïdaux proposée dans cette thèse est donc de tirer parti de cette propriété afin d'améliorer les performances du solveur tout en contrôlant l'erreur de calcul. Elle consiste à décomposer la solution en deux parties : une sinusoïde, dont les coefficients de Fourier sont fixés pour chaque intervalle d'intégration puis mis à jour par estimation paramétrique, et un terme de correction sur lequel le système d'EDA est reformulé et résolu à l'aide d'un schéma d'intégration à pas adaptatif. Une attention particulière a été portée au choix de l'estimateur paramétrique, ce dernier ayant un impact direct sur le pas d'intégration de par sa précision et indirect de par son effet sur la stabilité globale de la méthode. L'estimateur finalement développé consiste à calculer les coefficients de Fourier qui minimisent une mesure de la stationnarité du système. Ce dernier étant convergent en régime permanent, le terme de correction est progressivement amorti, permettant ainsi d'accroître considérablement le pas d'intégration. Cette méthode, intégrée au sein du solveur SUNDIALS IDA puis interfacée avec un moteur de calcul industriel, permet d'accélérer très nettement les simulations en comparaison avec une implémentation classique

Solving differential equations (PDEs/ODEs/DAEs) is central to the simulation of physical phenomena. The increase in size and complexity of the models requires the design of methods that are robust and efficient in terms of computational time. The aim of this thesis is to design methods that accelerate the solution of differential equations by domain decomposition methods. We first consider Schwarz domain decomposition methods to solve large-scale linear systems arising from the discretization of PDEs. In order to accelerate the convergence of the Schwarz method, we propose an approximation of the error propagation operator. This approximation preserves the structure of the exact operator. A significant reduction of computational time is obtained for the groundwater flow problem in highly heterogeneous media. The second contribution concerns solving the sequence of linear systems arising from the time-integration of nonlinear problems. We propose two approaches, taking advantage of the fact that the Jacobian matrix does not change dramatically from one system to another. First, we apply Broyden’s update to the Restricted Additive Schwarz (RAS) preconditioner instead of recomputing the local LU factorizations. The second approach consists of dedicating processors to the asynchronous and partial update of the RAS preconditioner. Numerical results for the lid-driven cavity problem, and for a reaction-diffusion problem show that a super-linear speedup may be achieved. The last contribution concerns the simultaneous solution of nonlinear problems associated to consecutive time steps. We study the case where the Broyden method is used to solve these nonlinear problems. In that case, Broyden’s update of the Jacobian matrix may also be propagated from one time step to another. The parallelization through the time steps is also applied to the problem of finding a consistent initial guess for differential-algebraic equations, La résolution des équations différentielles (EDP/EDO/EDA) est au cœur de la simulation de phénomènes physiques. L'accroissement de la taille et de la complexité des modèles nécessite la mise en œuvre de méthodes de résolution robustes et performantes en termes de temps de calcul. L'objectif de cette thèse est de proposer des méthodes pour accélérer la résolution des équations différentielles par des méthodes de décomposition de domaine. On considère d'abord les méthodes de décomposition de domaine de Schwarz pour la résolution de grands systèmes linéaires issus de la discrétisation d'EDP. Afin d'accélérer la convergence de la méthode de Schwarz, on propose une approximation de l'opérateur de propagation d'erreur. Cette approximation respectera la structure de l'opérateur exact, ce qui conduira à une réduction très significative des temps de calcul sur le problème des écoulements dans les milieux poreux hétérogènes. La deuxième contribution concerne la résolution de la suite de systèmes linéaires provenant de l'intégration en temps de problèmes non linéaires. On propose deux approches en utilisant le fait que la matrice jacobienne ne varie que peu d'un système à l'autre. Premièrement, on applique la mise à jour de Broyden au préconditionneur RAS (Restricted Additive Schwarz) au lieu de recalculer les factorisations LU. La deuxième approche consiste à dédier des processeurs a la mise à jour partielle et asynchrone du préconditionneur RAS. Des résultats numériques sur le problème de la cavité entrainée et sur un problème de réactiondiffusion montrent qu'une accélération super linéaire peut être obtenue. La dernière contribution a pour objet la résolution simultanée des problèmes non linéaires de pas de temps consécutifs. On étudie le cas où la méthode de Broyden est utilisée pour résoudre ces problèmes non linéaires. Dans ce cas, la mise à jour de Broyden peut être propagée d'un pas de temps à l'autre. La parallélisation à travers les pas de temps est également appliquée a la recherche d'une solution initiale consistante pour les équations différentielles algébriques

A two level preconditioner, based on the Aitken acceleration technique of a sequence of q interface’s solution vectors of the Restricted Additive Schwarz iterative process, is designed. This new technique, called ARAS(q), uses a coarse approximation of the solution on the interface. Different methods are discussed, leading to the development of an approximation technique by Singular Value De- composition of the sequence of vectors. Parallel implementations of Aitken-Schwarz methods are proposed, and the study leads to a fully algebraic one-level and two- level MPI implementation of ARAS(q) written into the PETSc library framework. This fully parallel and algebraic code gives an adaptive tool to solve linear systems such as those arising from automatic differentiation of compressible Navier-Stokes solution with respect to the simulation’s parameters, Un préconditionneur à deux niveaux, reposant sur la technique d’accélération d’Aitken d’une suite de q vecteurs solutions de l’interface d’un pro- cessus itératif de Schwarz Additif Restreint, est conçu. Cette nouvelle technique, dénomée ARAS(q), utilise une approximation grossière de la solution sur l’interface. Différentes méthodes sont proposées, aboutissant au développement d’une tech- nique d’approximation par Décomposition en Valeures Singulières de la suite de vecteurs. Des implémentations parallèles des méthodes d’Aitken-Schwarz sont pro- posées et l’étude conduit à l’implémentation d’un code totalement algébrique, sur un ou deux niveaux de parallélisation MPI, écrit dans l’environnement de la biblio- thèque PETSc. Cette implémentation pleinement parallèle et algébrique procure un outil flexible pour la résolution de systèmes linéaires tels que ceux issus de la dif- férentiation automatique des solutions de Navier-Stokes dépendant des paramètres de la simulation

Complexification of multi-physics modeling leads to have to simulate systems of ordinary differential equations and algebraic differential equations with increasingly large numbers of unknowns and over large times of simulation. In addition the evolution of parallel computing architectures requires other ways of parallelization than the decomposition of system in subsystems. In this work, we propose to design domain decomposition methods in time for the resolution of EDO. We reformulate the initial value problem in a boundary values problem on the symmetrized time interval, under the assumption of reversibility of the flow. We develop two methods, the first connected with a Schur complement method, the second based on a Schwarz type method for which we show convergence, being able to be accelerated by the Aitken method within the linear framework. In order to accelerate the convergence of the latter within the non-linear framework, we introduce the techniques of extrapolation and of acceleration of the convergence of non-linear sequences. We show the advantages and the limits of these techniques. The obtained results lead us to develop the acceleration of the method of the type Schwarz by a Newton method. Finally we investigate non-linear matching conditions adapted to the domain decomposition of nonlinear problems. We make use of the port-Hamiltonian formalism, resulting from the control field, to deduce the matching conditions in the framework of the shallow-water equation and the non-linear heat equation. After an analytical study of the convergence of the DDM associated with these conditions of transmission, we propose and study a formulation of augmented Lagrangian under the assumption of separability of the constraint., La complexification de la modélisation multi-physique conduit d’une part à devoir simuler des systèmes d’équations différentielles ordinaires et d’équations différentielles algébriques de plus en plus grands en nombre d’inconnues et sur des temps de simulation longs. D’autre part l’évolution des architectures de calcul parallèle nécessite d’autres voies de parallélisation que la décomposition de système en sous-systèmes. Dans ce travail, nous proposons de concevoir des méthodes de décomposition de domaine pour la résolution d’EDO en temps. Nous reformulons le problème à valeur initiale en un problème aux valeurs frontières sur l’intervalle de temps symétrisé, sous l’hypothèse de réversibilité du flot. Nous développons deux méthodes, la première apparentée à une méthode de complément de Schur, la seconde basée sur une méthode de type Schwarz dont nous montrons la convergence pouvant être accélérée par la méthode d’Aitken dans le cadre linéaire. Afin d’accélérer la convergence de cette dernière dans le cadre non-linéaire, nous introduisons les techniques d’extrapolation et d’accélération de la convergence des suites non-linéaires. Nous montrons les avantages et les limites de ces techniques. Les résultats obtenus nous conduisent à développer l’accélération de la méthode de type Schwarz par une méthode de Newton. Enfin nous nous intéressons à l’étude de conditions de raccord non-linéaires adaptées à la décomposition de domaine de problèmes non-linéaires. Nous nous servons du formalisme hamiltonien à ports, issu du domaine de l’automatique, pour déduire les conditions de raccord dans le cadre l’équation de Saint-Venant et de l’équation de la chaleur non-linéaire. Après une étude analytique de la convergence de la DDM associée à ces conditions de transmission, nous proposons et étudions une formulation de Lagrangien augmenté sous l’hypothèse de séparabilité de la contrainte.

Ce rapport présente le travail effectué au sein de l’UMR EMMAH à Avignon sur le code éléments finis Fafemo. Ce code donne une solution numérique des équations non linéaires de Richards traduisant les écoulements dans les milieux poreux non saturés. J’ai d’abord amélioré le code séquentiel en utilisant des méthodes itératives pour résoudre les systèmes linéaires creux (PETSc) et le mailleur Gmsh. Les coûts de calcul ont été significativement réduis. Cependant, l’objectif principal de ce stage a été de concevoir une version parallèle et efficace de Fafemo. J’ai effectué les tests de cette version parallèle sur deux clusters (Migale et Jade, le plus puissant cluster français). Ce travail a permis de faire un premier pas vers la modélisation du comportement de milieux poreux fortement hétérogènes comme les substrats des toitures terrasses végétalisées, This manuscrit presents the work done on Fafemo software based on finite element methods at the EMMAH laboratory in Avignon. This software gives us numerical solutions of Richards non linear equations for water flow in unsaturated porous media. First I improved the software using iterative methods to solve sparse linear systems (PETSc) and Gmsh as mesher. Computational costs have been significantly reduced. But the main goal of this training course was to develop a parallel version of Fafemo. This version was tested on Migale and Jade (the most powerful French cluster). This works leads to a first step in simulating water flow in heterogeneous ground with application to green roofs

This PhD Thesis deals with the development of parallel numerical methods for solving Ordinary and Algebraic Differential Equations. ODE and DAE are commonly arising when modeling complex dynamical phenomena. We first show that the parallelization across the method is limited by the number of stages of the RK method or DIMSIM. We introduce the Schur complement into the linearised linear system of time integrators. An automatic framework is given to build a mask defining the relationships between the variables. Then the Schur complement is coupled with Jacobian Free Newton-Krylov methods. As time decomposition, global time steps resolutions can be solved by parallel nonlinear solvers (such as fixed point, Newton and Steffensen acceleration). Two steps time decomposition (Parareal, Pita,...) are developed with a new definition of their grids to solved stiff problems. Global error estimates, especially the Richardson extrapolation, are used to compute a good approximation for the second grid. Finally we propose a parallel deferred correction, La simulation numérique de systèmes d’équations différentielles raides ordinaires ou algébriques est devenue partie intégrante dans le processus de conception des systèmes mécaniques à dynamiques complexes. L’objet de ce travail est de développer des méthodes numériques pour réduire les temps de calcul par le parallélisme en suivant deux axes : interne à l’intégrateur numérique, et au niveau de la décomposition de l’intervalle de temps. Nous montrons l’efficacité limitée au nombre d’étapes de la parallélisation à travers les méthodes de Runge-Kutta et DIMSIM. Nous développons alors une méthodologie pour appliquer le complément de Schur sur le système linéarisé intervenant dans les intégrateurs par l’introduction d’un masque de dépendance construit automatiquement lors de la mise en équations du modèle. Finalement, nous étendons le complément de Schur aux méthodes de type "Krylov Matrix Free". La décomposition en temps est d’abord vue par la résolution globale des pas de temps dont nous traitons la parallélisation du solveur non-linéaire (point fixe, Newton-Krylov et accélération de Steffensen). Nous introduisons les méthodes de tirs à deux niveaux, comme Parareal et Pita dont nous redéfinissons les finesses de grilles pour résoudre les problèmes raides pour lesquels leur efficacité parallèle est limitée. Les estimateurs de l’erreur globale, nous permettent de construire une extension parallèle de l’extrapolation de Richardson pour remplacer le premier niveau de calcul. Et nous proposons une parallélisation de la méthode de correction du résidu.