UNIVERSIT E´ PARIS 13 No attribu´epar la biblioth`eque TH ESE` pour obtenir le grade de DOCTEUR DE L’UNIVERSIT E´ PARIS 13 Discipline: Math´ematiques Appliqu´ees Laboratoire d’accueil: ONERA - Le centre fran¸cais de recherche a´erospatiale Pr´esent´ee et soutenue publiquement le 19 d´ecembre 2014 par Oana Alexandra CIOBANU Titre M´ethode de d´ecomposition de domaine avec adaptation de maillage en espace-temps pour les ´equations d’Euler et de Navier–Stokes devant le jury compos´ede: Fran¸cois Dubois Rapporteur Laurence Halpern Directrice de th`ese Rapha`ele Herbin Rapporteure Xavier Juvigny Examinateur Olivier Lafitte Examinateur Juliette Ryan Encadrante UNIVERSITY PARIS 13 THESIS Presented for the degree of DOCTEUR DE L’UNIVERSIT E´ PARIS 13 In Applied Mathematics Hosting laboratory: ONERA - The French Aerospace Lab presented for public discussion on 19 decembre 2014 by Oana Alexandra CIOBANU Subject Adaptive Space-Time Domain Decomposition Methods for Euler and Navier–Stokes Equations Jury: Fran¸cois Dubois Reviewer Laurence Halpern Supervisor Rapha`ele Herbin Reviewer Xavier Juvigny Examiner Olivier Lafitte Examiner Juliette Ryan Supervisor newpage Remerciements Tout d’abord, je remercie grandement Juliette Ryan pour avoir accept´e d’ˆetre mon encad- rante de stage puis mon encadrante de th`ese. Pendant plus de trois ans, elle m’a fait d´ecouvrir mon m´etier de jeune chercheuse avec beaucoup de patience et de professionnalisme. Elle m’a soutenue, elle a ´et´ed’une disponibilit´eet d’une ´ecoute extraordinaires, tout en sachant ˆetre rigoureuse et exigeante avec moi comme avec elle-mˆeme. Humainement, j’ai beaucoup appr´eci´e la relation d’amiti´eque nous avons entretenue, le climat de confiance que nous avons maintenu et les discussions extra-math´ematiques que nous avons pu avoir et qui ont renforc´ele lien que nous avions. Je la remercie pour tout ¸ca, et je sais que j’en oublie. Je remercie sinc`erement Rapha`ele Herbin et Fran¸cois Dubois d’avoir accept´ela lourde tˆache d’ˆetre mes rapporteurs, et je les remercie pour l’attention qu’ils ont port´e`amon travail dans un d´elai plus que raisonnable. C’est un plaisir et un honneur pour moi d’avoir dans mon jury Olivier Lafitte. C’est grˆace `alui que je suis arriv´ee en MACS, et je suis tr`es fi`ere de le retrouver pour cette soutenance de th`ese, c’est une jolie mani`ere de boucler la boucle. Je remercie ´egalement mes coll`egues de l’Onera, pour les nombreuses discussions que nous avons eues au cours de ces trois ann´ees et pour les moments sympathiques que nous avons pass´es au laboratoire. En particulier je voudrais remercier Xavier Juvigny, tout d’abord parce qu’il a suivi de tr`es pr`es l’avanc´ee de mon travail, et surtout pour sa disponibilit´eet ses facult´es pour me d´epatouiller de tous mes probl`emes informatiques. J’ai eu ´enorm´ement de plaisir `ale cˆotoyer durant ces trois ann´ees, `a´echanger avec lui sur la th`ese, sur la vie, et sur plein de choses en g´en´eral. Sans lui, la th`ese n’aurait pas ´et´edrˆole du tout et je suis vraiment contente d’avoir fait sa connaissance et de m’ˆetre li´ed’amiti´eavec lui. Je vais `apr´esent remercier mes proches. Je remercie mes parents et mes fr`eres pour leur soutien et la confiance aveugle qu’ils ont eue envers ma r´eussite. Je remercie Mikael, Roxana, Raluca, Luciana, Diana et Alina pour leur soutien moral. Je remercie Roxana pour avoir relu une partie de ma th`ese et pour leurs corrections, Alina pour l’aide qu’elle m’a apport´ee pour pr´eparer le pot, et Mikael pour les deux. Je tiens `aexprimer ma profonde reconnaissance `aLaurence Halpern, qui m’a fait l’honneur d’ˆetre ma directrice de th`ese et pour toutes les discussions constructives et conseils autour mon travail. Ma th`ese ne se serait sans doute jamais achev´ee sans leur soutien et je les remercie pour tous les moments qu’on a pass´es ensemble. Contents Introduction iii 1 Numerical Methods for Euler and Navier–Stokes Equations 1 1.1 FluidDynamicsEquations.............................. 2 1.1.1 Compressible Navier–Stokes Equations . 4 1.1.2 EulerEquations ................................ 5 1.2 FiniteVolume(FV)Method............................. 5 1.2.1 Space discretisation . 7 1.2.2 Space-Time Boundary Conditions . 8 1.2.3 Development of the numerical Euler flux . 16 1.2.4 Development of the numerical viscous flux . 24 1.3 ExplicitandImplicitTimeSchemes . 25 1.3.1 First and Second order Explicit Methods . 26 1.3.2 Second order Implicit Backward Differentiation Methods . 28 2 Schwarz based Domain Decomposition Methods (DDMs) 39 2.1 Classical Schwarz domain decomposition methods . 41 2.1.1 Alternating Schwarz Algorithm . 42 2.1.2 Parallel Schwarz Algorithm . 43 2.2 Schwarz Waveform Relaxation (SWR) Method . 45 2.3 Adaptive Schwarz Waveform Relaxation (ASWR) Method . 47 2.4 SchwarzMethodsappliedtoimplicitsolvers . 50 2.5 Transmission conditions on artificial boundaries . 51 2.5.1 Dirichlet type boundary conditions . 54 2.5.2 Mixed Dirichlet/Robin boundary conditions . 55 2.5.3 Robin (Fourier) Boundary Condition . 58 2.6 Convergence and Stopping criteria . 61 3 CFD Code organisation and description 71 3.1 Code description and programming techniques . 73 3.2 Parallelisation techniques . 77 3.2.1 Parallel computing inside loops (OpenMP) . 78 3.2.2 Parallel computing via message passing (MPI) . 80 3.2.3 Graphic Processor Unit (GPU) . 81 3.3 Parallelism efficiency evaluation . 87 4 Numerical results and discussions 91 4.1 Comparisons of numerical convective fluxes schemes . 92 4.1.1 1Dshocktubeproblem ............................ 92 4.1.2 2D Forward Facing Step . 98 4.2 Applications of Domain Decomposition techniques . 105 i ii Contents 4.2.1 GPUversusCPU................................105 4.2.2 Exact solution for Euler equations: 2D isentropic vortex . 106 4.2.3 Sound generation in a 2D low-Reynolds mixing layer . 113 4.3 Vortex shedding from rectangles . 117 Conclusion and Perspectives 129 A Computation of diffusive Jacobians 133 B List of GPU libraries 141 C Example of CUDA programming. Minimum reduction 143 Nomenclature 149 Bibliography 151 Introduction This work focuses on the research field of laminar flow of an ideal gas around solid bodies, on the resolution of aerodynamic multi-scale problems that are costly and difficult to solve in their original form. In the aerodynamic field, scientists are facing many different problems, geometrical problems due to the complex design of civil aircraft, military aircraft, helicopters, space transport, missile systems, launchers, air intakes, nozzles, propulsive jets and other bodies in movement. Other problems are issued from the model used to describe the physical configu- ration. Most of the models are robust only for particular cases, coefficients are established on particular cases as well, for simplified models and small range data problems. Thus, phenomena too complex to be reliably predicted by theory and too dangerous or expensive to be reproduced in the laboratories must be simulated by computational scientists. A simulation with low range precisions can lead to false results, unrealistic behaviour and even damages if used in experi- mental work. At all levels, in order to better approach the geometry and to better validate a model, for physical viability precision, large data systems must be solved. Several techniques of parallel computing have been developed as they allow to solve large data systems of equa- tions. They essentially depend on the chosen domain decomposition method, but convergence problems may occur for large numbers of sub-domains. In what concerns the numerical point of view on solving the Navier–Stokes system of equa- tions, robust and fast methods are now available, which combine non-linear and linear solvers requiring less memory capacity. In the context of long term simulations, the class of global implicit approaches has proved its superiority as they are able to simulate a quasi-steady-state behaviour without being restricted to short time steps to ensure convergence. However, in in- dustrial applications explicit methods are preferred, as they are easy to compute and to adapt, but very expensive since they are restricted to a small time step. Between the range of existing algorithms the choice of an appropriate one is still uncertain. The performance of both, explicit and implicit methods can be improved when combined with a domain decomposition method. Domain decomposition methods split large problems into smaller sub-problems that can be solved in parallel. Space domain decomposition method is used to provide high-performing algorithms in many fields of numerical applications. Yet, the technology evolution towards multi-cores, multi-processors, GPU and multi-GPUs is not completely exploited. To achieve full performance on large clusters with up to 100 000 nodes (such as recently the IBM Sequoia, or GPUs) the time dimension has to be taken into account. An essential gain to be obtained from time-space domain decomposition is the ability to apply different time-space discretisations on sub-domains thus improving efficiency and convergence of implicit schemes. The idea of domain decomposition is quite old, it was proposed for the first time in 1869 by H.A. Schwarz [87] with the purpose of solving a linear problem (Laplace equation) over a geometry, which it was impossible to solve in its initial form. Schwarz splits the global problem into two overlapping problems and solves them alternately by exchanging interface conditions. He manages in this way to solve for the first time the Laplace equation for a complex geometry and gives the main goal to the domain decomposition method, the one of managing complicated geometries.