Nonoverlapping Domain Decomposition Applied to a Computational Fluid Mechanics Code
The purpose of this paper is the description of the development and implementation of the linear part of a numerical algorithm for the simulation of a Newtonian fluid flow and the parallelization of that code on several computer architectures. The test pr
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Abstract. The purpose of this paper is the description of the development and implementation of the linear part of a numerical algorithm for the simulation of a Newtonian fluid flow and the parallelization of that code on several computer architectures. The test problem treated is the steady state, laminar, incompressible, isothermic, 2D fluid flow (extendible to 3D case), the Navier-Stokes equations being discretized by a fully coupled finite volume method. For this problem, sparse data structures, nonstationary iterative methods and several preconditioners are applied. The numerical results allow the conclusion that the fully coupled version can compete with the decoupled classic SIMPLE method (SemiImplicit Method for Pressure-Linked Equations), by using the Krylov subspace methods. Parallel versions of the coupled method based on nonoverlapping domain decomposition are discussed.
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Problem Specification and Brief Description of the Algorithms
The test problem used is the steady state two-dimensional incompressible laminar flow in a square lid-driven cavity. This academic problem describes the nonlinear and elliptic behavior of many flows studied in engineering and it is used to test the efficiency of many algorithms [6]. The Navier-Stokes equations governing the flow in this case are ∂ (Ui ) = 0 ∂xi ∂P ∂ ∂ (Ui Uj ) = − + (τij ) , ρ ∂xj ∂xi ∂xj
(1) τij = µ
∂Ui ∂Uj + ∂xj ∂xi
(2)
for i=1,2 (in 2-D case) and i=1,2,3 (in 3-D case), respectively, the continuity equation (mass conservation) and the momentum equations, where ρ stands for the density, µ for the dynamic viscosity, Re for the Reynolds number, U for the velocity field where Ui for the velocity along xi direction (i = 1, 2 for the 2-D case), and P for the static pressure. The algorithms reported here were also applied to other related mechanical problems, mainly the laminar backward-facing step flow, with similar behaviour. P. Amestoy et al. (Eds.): Euro-Par’99, LNCS 1685, pp. 608–612, 1999. c Springer-Verlag Berlin Heidelberg 1999
Nonoverlapping Domain Decomposition
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The classical SIMPLE [3] algorithm solves in a decoupled way the governing differential equations. The resulting linear systems, one for each velocity and pressure, are solved by TDMA which is derived from Gaussian elimination adapted to the particular case of tridiagonal systems. The DIRECTO [1], solves the fluid flow equations as a completely coupled system. The resulting square coefficient matrix is large, sparse, block tridiagonal and unsymmetric. The matrix size is n×n, n = 3×ni×nj for a ni×nj grid. So at each stage of the DIRECTO procedure such a linear system is to be solved and this is the most time consuming part of the algorithm. Another important key of the method is that there are no null entries on the diagonal of the coefficient matrix due to the existence of pressure terms on the continuity equation. For more details about relations to other similar methods see [6] [7] and references in them and for the accuracy of DIRECTO see [2]. Domain decomposition techniques
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