The Krylov accelerated SIMPLE(R) method for flow problems in industrial furnaces

Abstract. This paper discusses the solution of the Stokes problem using block preconditioned iterative methods. Block preconditioners are based on the block factorization of the discretized problem. We focus on two specific types: SIMPLE-type preconditioners and the LSC preconditioner. Both methods use scaling to improve their performance. We test convergence of GCR in combination with these preconditioners both for a constant and a non-constant viscosity Stokes problem.

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“…In this section, we discuss variants of the SIMPLE preconditioner [8]. The SIM-PLE preconditioner can be written as…”

Section: Simple-type Preconditionersmentioning

confidence: 99%

Block Preconditioners for the Incompressible Stokes Problem

Rehman

Vuik

Segal

2010

Large-Scale Scientific Computing

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“…D is the diagonal of the convection diffusion matrix andŜ = −BD −1 B T , an approximation to the Schur complement. Vuik et al [16], used SIMPLE and its variants as a preconditioner to solve the incompressible Navier-Stokes problem. One iteration of the SIMPLE algorithm with assumption p * = 0 is used as a preconditioner.…”

Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

“…The steps given in (16) and (17) contain two Poisson solves, two velocity subproblems solves-posed to one velocity solve in the classical algorithm-and matrix vector updates. However, in practical SIMPLER implementation -per preconditioning step -the velocity solve is performed once because no significant effect on the convergence with the SIMPLER preconditioner is observed, see Appendix-A.…”

Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

“…In the remainder of this paper, we will use SIMPLER with one velocity solve. One iteration of the SIMPLER algorithm is approximately 1.3 times more expensive than the SIMPLE iteration [16]. SIMPLER convergence is usually faster than SIMPLE.…”

Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

“…The preconditioners are based on the knowledge of the pressure and velocity matrices. Examples of such preconditioners can be found in [10][11][12][13][14][15][16]. The goal of all these preconditioners is to get convergence independent of the Reynolds number and grid size.…”

Section: Introductionmentioning

confidence: 99%

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SIMPLE‐type preconditioners for the Oseen problem

Rehman

Vuik

Segal

2008

Numerical Methods in Fluids

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In this paper, the steady incompressible Navier–Stokes equations are discretized by the finite element method. The resulting systems of equations are solved by preconditioned Krylov subspace methods. Some new preconditioning strategies, both algebraic and problem dependent are discussed. We emphasize on the approximation of the Schur complement as used in semi implicit method for pressure‐linked equations‐type preconditioners. In the usual formulation, the Schur complement matrix and updates use scaling with the diagonal of the convection–diffusion matrix. We propose a variant of the SIMPLER preconditioner. Instead of using the diagonal of the convection–diffusion matrix, we scale the Schur complement and updates with the diagonal of the velocity mass matrix. This variant is called modified SIMPLER (MSIMPLER). With the new approximation, we observe a drastic improvement in convergence for large problems. MSIMPLER shows better convergence than the well‐known least‐squares commutator preconditioner which is also based on the diagonal of the velocity mass matrix. Copyright © 2008 John Wiley & Sons, Ltd.

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A fractional two‐step implicit algorithm using curvilinear collected grids

Ye¹,

McCorquodale

Peng

2006

Numerical Methods in Fluids

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SUMMARYAn e cient fractional two-step implicit algorithm is reported to simulate incompressible uid ows in a boundary-ÿtted curvilinear collocated grid system. Using the ÿnite volume method, the convection terms are discretized by the high-accuracy Roe's scheme to minimize numerical di usion. An implicitness coe cient is introduced to accelerate the rate of convergence. It is demonstrated that the proposed algorithm links the fractional step method to the pressure correction procedure, and the SIMPLEC method could be considered as a special case of the fractional two-step implicit algorithm (when = 1). The proposed algorithm is applicable to unsteady ows and steady ows. Three benchmark twodimensional laminar ows are tested to evaluate the performance of the proposed algorithm. Performance is measured by sensitivity analyses of the e ciency, accuracy, grid density, grid skewness and Reynolds number on the solutions. Results show that the model is e cient and robust.

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The Krylov accelerated SIMPLE(R) method for flow problems in industrial furnaces

Cited by 40 publications

References 24 publications

Block Preconditioners for the Incompressible Stokes Problem

Block Preconditioners for the Incompressible Stokes Problem

SIMPLE‐type preconditioners for the Oseen problem

A fractional two‐step implicit algorithm using curvilinear collected grids

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