Efficient and scalable discretization of the Navier–Stokes equations with LPS modeling

Abstract

In this work, we address the solution of the Navier-Stokes equations (NSE) by a Finite Element (FE) Local Projection Stabilization (LPS) method. The focus is on a LPS method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure, which only acts on the high frequency components of the flow. As a main contribution, we propose and test an efficient discretization of the model via a stable velocity-pressure segregation, using semi-implicit Backward Differentiation Formulas (BDF) in time. On the one hand, numerical studies illustrate that the solver accurately reproduces first and second-order statistics of benchmark turbulent flows for relatively coarse meshes. On the other hand, they show that the solver works in an efficient (i.e., robust and fast) way, especially when interfaced with scalable domain decomposition methods. Such scalability results are obtained on up to 16,384 cores with a near-ideal speedup.

Assessment of self-adapting local projection-based solvers for laminar and turbulent industrial flows

“…Nevertheless, the number of GMRES iterations remains stable, and in the same low range as in Ref. [7] (see Table 1). To sum up, also the parallel performances in the case of equal-order FE are rather satisfactory, and seem to be in accordance with the current state-of-the-art, e.g., [28].…”

“…This is reflected in Fig. 6, where we start to observe from 4096 processes that using equal-order FE leads to a slight deterioration in the scalability of the total average time to complete a time step, which is not the case when considering mixed FE in [7]. This follows from the fact that using equal-order FE requires the assembly of an additional term for pressure stabilization and, as consequence, this results in an increased computational cost with respect to mixed formulations.…”

“…Anyway, as displayed in Table 1, the preconditioners are both numerically extremely stable, with numbers of iterations remaining in the same low range. The GMRES method is stopped when the relative preconditioner residual is lower than 10 -8 for the velocity unknows and 10 -6 for the pressure unknows Remark 3.2 Note that this numerical study differs from the one performed in [7], where just mixed inf-sup stable FE of Taylor-Hood type (P 2 /P 1 ) have been considered for the pair velocity/pressure in numerical experiments, for which the pressure stabilized term (7) is neglected. The presented results with equal-order P 2 /P 2 FE considering pressure stabilization are almost comparable with the ones of [7], thus being in good agreement with experimental data.…”

“…Solving the associated large linear systems could become extremely expensive from the computational point of view, that is why we adopt in the numerical implementation a highly parallel strategy based on Domain Decomposition Methods (DDM). Both steps are solved by using a DDM preconditioner with the GMRES iterative method applied to the associated system in the parallel framework described in [7], and a convincing strong scaling analysis of the used algorithm is showcased in this framework. In particular, we have interfaced the proposed fully discrete scheme (17)- (18) with HPDDM [20,21], a high performance unified framework for DDM, and used a parallel iterative linear solver based on an optimized Schwarz DDM as preconditioner [20,22].…”

“…We will also focus on an efficient time discretization of this method via a stable velocity-pressure segregation, using semi-implicit Backward Differentiation Formulas up to the second order (BDF2), with a special emphasis on its numerical solution in a parallel setting (cf. [7]). We show some relevant 3D numerical tests, to assess the performance of the proposed LPS method as an efficient and accurate solver for the simulation of laminar and turbulent incompressible complex flows that could arise in industrial applications.…”