Parallel Iterative Methods for Navier-Stokes Equations and Application to Stability Assessment (Distinguished Paper) (original) (raw)
Related papers
2010
Based on full domain partition, three parallel iterative finite element algorithms for the stationary Navier-Stokes equations are proposed and analyzed. In these algorithms, each subproblem is defined in the entire domain with the vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Under some (strong) uniqueness conditions, errors of the parallel iterative finite element solutions are estimated. Some numerical results are also given which demonstrate the efficiency of the parallel iterative algorithms.
Scalable Block Preconditioners for Linearized Navier-Stokes Equations at High Reynolds Number
Algorithms
We review a number of preconditioners for the advection-diffusion operator and for the Schur complement matrix, which, in turn, constitute the building blocks for Constraint and Triangular Preconditioners to accelerate the iterative solution of the discretized and linearized Navier-Stokes equations. An intensive numerical testing is performed onto the driven cavity problem with low values of the viscosity coefficient. We devise an efficient multigrid preconditioner for the advection-diffusion matrix, which, combined with the commuted BFBt Schur complement approximation, and inserted in a 2×2 block preconditioner, provides convergence of the Generalized Minimal Residual (GMRES) method in a number of iteration independent of the meshsize for the lowest values of the viscosity parameter. The low-rank acceleration of such preconditioner is also investigated, showing its great potential.
A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number
Journal of Scientific Computing, 2013
Direct numerical simulation of three-dimensional incompressible flows at high Reynolds number using the unsteady Navier-Stokes equations is challenging. In order to obtain accurate simulations, very fine meshes are necessary, and such simulations are increasingly important for modern engineering practices, such as understanding the flow behavior around high speed trains, which is the target application of this research. To avoid the time step size constraint imposed by the CFL number and the fine spacial mesh size, we investigate some fully implicit methods, and focus on how to solve the large nonlinear system of equations at each time step on large scale parallel computers. In most of the existing implicit Navier-Stokes solvers, segregated velocity and pressure treatment is employed. In this paper, we focus on the Newton-Krylov-Schwarz method for solving the monolithic nonlinear system arising from the fully coupled finite element discretization of the Navier-Stokes equations on unstructured meshes. In the subdomain, LU or point-block ILU is used as the local solver. We test the algorithm for some threedimensional complex unsteady flows, including flows passing a high speed train, on a supercomputer with thousands of processors. Numerical experiments show that the algorithm has superlinear scalability with over three thousand processors for problems with tens of millions of unknowns.
Block preconditioning and domain decomposition methods. II
Journal of Computational and Applied Mathematics, 1988
Domain decomposition methods for the solution of partial differential equations are attractive on parallel processors, because each processor can work independently on a large subtask. The corresponding stiffness matrix takes a sparse block structure, for which preconditioned iterative methods can be used when solving linear systems with the stiffness matrix. For domains decomposed in strips we get a blocktridiagonal structure for which a new block LU preconditioner was presented in an earlier report [5] by the authors. An alternative method, and also the one more commonly used for substructuring methods, is based on approximation of the Schur complement matrix. This approximation is frequently done by various difference methods (see [6], [ll], and [16]). In the present paper we examine methods based on algebraic approximation methods. This is similar to methods used by Chan [9].
Preconditioners for Incompressible Navier-Stokes Solvers
Numerical Mathematics: Theory, Methods and Applications, 2010
In this paper we give an overview of the present state of fast solvers for the solution of the incompressible Navier-Stokes equations discretized by the finite element method and linearized by Newton or Picard's method. It is shown that block preconditioners form an excellent approach for the solution, however if the grids are not to fine preconditioning with a Saddle point ILU matrix (SILU) may be an attractive alternative. The applicability of all methods to stabilized elements is investigated. In case of the stand-alone Stokes equations special preconditioners increase the efficiency considerably.
Computational Fluid Dynamics - Basic Instruments and Applications in Science, 2018
Implicit methods based on the Newton's rootfinding algorithm are receiving an increasing attention for the solution of complex Computational Fluid Dynamics (CFD) applications due to their potential to converge in a very small number of iterations. This approach requires fast convergence acceleration techniques in order to compete with other conventional solvers, such as those based on artificial dissipation or upwind schemes, in terms of CPU time. In this chapter, we describe a multilevel variable-block Schur-complement-based preconditioning for the implicit solution of the Reynoldsaveraged Navier-Stokes equations using unstructured grids on distributed-memory parallel computers. The proposed solver detects automatically exact or approximate dense structures in the linear system arising from the discretization, and exploits this information to enhance the robustness and improve the scalability of the block factorization. A complete study of the numerical and parallel performance of the solver is presented for the analysis of turbulent Navier-Stokes equations on a suite of threedimensional test cases.
Domain decomposition methods in computational fluid dynamics
International Journal for Numerical Methods in Fluids, 1992
The divide-and-conquer paradigm of iterative domain decomposition, or substructuring, has become a practical tool in computational fluid dynamics applications because of its flexibility in accommodating adaptive refinement through locally uniform (or quasi-uniform) grids, its ability to exploit multiple discretizations of the operator equations, and the modular pathway it provides towards parallelism. We illustrate these features on the classic model problem of flow over a backstep using Newton's method as the nonlinear iteration. Multiple discretizations (second-order in the operator and first-order in the preconditioner) and locally uniform mesh refinement pay dividends separately_ and they can be combined synergistically. We include sample performance results from an Intel iPSC/860 hypercube implementation.