Développement et validation d’un solveur linéaire hybride RSVD-SOR pour les équations de Navier-Stokes 3D incompressibles

Abstract

Incompressible Navier-Stokes equations can be solved using various numerical strategies, including projection methods. These approaches require the repeated resolution of a Poisson equation for pressure to satisfy the divergence-free constraint. This elliptic operation generally constitutes the primary computational cost of the simulations. The NSMP code used in this project is no exception to this constraint, as the resolution of the Poisson problem represents the dominant bottleneck. This thesis presents the development and validation of a hybrid linear solver designed to improve this resolution. The proposed approach combines Randomized Singular Value Decomposition (RSVD) with a Successive Over-Relaxation (SOR) smoother. The RSVD algorithm is used to quickly generate a low-rank approximation of the system, providing a robust initial guess that captures the dominant modes of the pressure field. The SOR smoother then intervenes to eliminate the remaining residuals and ensure convergence toward the exact solution of the original system. The accuracy of the method was validated using the Taylor Green vortex analytical test case, demonstrating a spatial convergence order of 2. Performance was evaluated through a strong scaling study ranging from 1 to 256 processors on the Narval supercomputer. The results demonstrate that by adjusting the energy threshold, the hybrid solver significantly outperforms the classic MultiSOR algorithm. The parametric flexibility of this algorithm allows for computational time optimization without compromising numerical rigor.

Date	15 May 2026
Original language	French
Awarding Institution	École de technologie supérieure
Supervisor	Camille Coti (Supervisor) & Damien Pham Van Bang (Co-supervisor)