Preconditioned conjugate gradient methods for the incompressible Navier‐Stokes equations

Chin, P.; D'Azevedo, Eduardo F.; Forsyth, Peter; Tang, Wei-Pai

doi:10.1002/fld.1650150303

Cited by 31 publications

(13 citation statements)

References 26 publications

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“…Here we discuss only the method's application to (1) and (2). Representing the convective derivative by D/Dt = d/dt + u .…”

Section: The Lagrange-galerkin Methodsmentioning

confidence: 99%

“…Applying the Lagrange-Galerkin method to (1) and (2) in the context of a mixed finite element approximation removes the non-linearity introduced by the advection term and gives rise to a symmetric and indefinite linear system of the form at each time step. Here A is N u x N u and symmetric positive definite, B is N, x Nu, 0 is the N, x N, zero matrix and the vectors u and p contain the N , nodal velocity and N, nodal pressure values respectively.…”

Section: Re U + U~v U + V P --V~u = O mentioning

confidence: 99%

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Iterative solution techniques for the stokes and Navier‐Stokes equations

Ramage

Wathen

1994

Numerical Methods in Fluids

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SUMMARYThe linear system arising from a Lagrange-Galerkin mixed finite element approximation of the NavierStokes and continuity equations is symmetric indefinite and has the same block structure as a system arising from a mixed finite element discretization of a Stokes problem. This paper considers the iterative solution of such a system, comparing the performance of the one-level preconditioned conjugate residual method for indefinite matrices with that of a more traditional two-level pressure correction approach. Asymptotic estimates for the amount of work involved in each method are given together with the results of related numerical experiments.

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“…Here we discuss only the method's application to (1) and (2). Representing the convective derivative by D/Dt = d/dt + u .…”

Section: The Lagrange-galerkin Methodsmentioning

confidence: 99%

Section: Re U + U~v U + V P --V~u = O mentioning

confidence: 99%

Iterative solution techniques for the stokes and Navier‐Stokes equations

Ramage

Wathen

1994

Numerical Methods in Fluids

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“…The imaginary part of w determines the rate of growth or decay of the Fourier mode, while the real part of w / k determines the wave speed c. The wave speed therefore vanishes for a < a0 and is given by c = J(a2 -a;) (18) for a > ao. Thus c is always less than a, becoming smaller for larger ao.…”

Section: Fourier Dispersion and Convergence Analysismentioning

confidence: 97%

Damped artificial compressibility iteration scheme for implicit calculations of unsteady incompressible flow

McHugh

Ramshaw

1995

Numerical Methods in Fluids

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SUMMARYPeyret (1 Fluid Mech., 7 8 , 4 9 4 3 (1976)) and others have described artificial compressibility iteration schemes for solving implicit time discretizations of the unsteady incompressible Navier-Stokes equations. Such schemes solve the implicit equations by introducing derivatives with respect to a pseudo-time variable z and marching out to a steady state in 2. The pseudo-time evolution equation for the pressurep takes the form ap/a7 = -a*V.u, where a is an artificial compressibility parameter and u is the fluid velocity vector. We present a new scheme of this type in which convergence is accelerated by a new procedure for setting a and by introducing an artificial bulk viscosity b into the momentum equation. This scheme is used to solve the non-linear equations resulting from a fully implicit time differencing scheme for unsteady incompressible flow. We find that the best values of a and b are generally quite different from those in the analogous scheme for steady flow (J. D. Ramshaw and V A. Mousseau, Comput. Fluids, 18, 361-367 (1990)), owing to the previously unrecognized fact that the character of the system is profoundly altered by the presence of the physical time derivative terms. In particular, a Fourier dispersion analysis shows that a no longer has the significance of a wave speed for finite values of the physical time step At. Indeed, if one sets a N 1u1 as usual, the artificial sound waves cease to exist when At is small and this adversely affects the iteration convergence rate. Approximate analytical expressions for a and b are proposed and the benefits of their use relative to the conventional values a N (u[ and b = 0 are illustrated in simple test calculations.

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“…In parallel (or vector) processing, ordering again plays a crucial rule. A number of studies have examined the effect of matrix ordering on the quality of preconditioners for iterative methods based on an incomplete factorization [8], 14], 16], [23], [24], [33], [21]. In [14], [16], and [17] evidence was presented to demonstrate that matrix ordering can have a profound effect on the quality of preconditioners.…”

Section: 1mentioning

confidence: 98%

Performance Issues for Iterative Solvers in Device Simulation

Qing¹,

McMacken²,

Tang³

1996

SIAM J. Sci. Comput.

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Due to memory limitations, iterative methods have become the method of choice for large scale semiconductor device simulation. However, it is well known that these methods suffer from reliability problems. The linear systems that appear in numerical simulation of semiconductor devices are notoriously ill conditioned. In order to produce robust algorithms for practical problems, careful attention must be given to many implementation issues. This paper concentrates on strategies for developing robust preconditioners. In addition, effective data structures and convergence check issues are also discussed. These algorithms are compared with a standard direct sparse matrix solver on a variety of problems.

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Preconditioned conjugate gradient methods for the incompressible Navier‐Stokes equations

Cited by 31 publications

References 26 publications

Iterative solution techniques for the stokes and Navier‐Stokes equations

Iterative solution techniques for the stokes and Navier‐Stokes equations

Damped artificial compressibility iteration scheme for implicit calculations of unsteady incompressible flow

Performance Issues for Iterative Solvers in Device Simulation

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