Krylov Methods for the Incompressible Navier-Stokes Equations

Edwards, W. Stuart; Tuckerman, Laurette S.; Friesner, Richard A.; Sorensen, Danny C.

doi:10.1006/jcph.1994.1007

Cited by 189 publications

(167 citation statements)

References 42 publications

(64 reference statements)

Supporting

Mentioning

165

Contrasting

Unclassified

Order By: Relevance

“…(1) comparison of values of critical Rayleigh numbers and neutral curves; (2) comparison of values of the aspect ratio where most dangerous perturbations replace each other; (3) comparison of patterns of the most dangerous perturbations.…”

Section: Discussionmentioning

confidence: 99%

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Gelfgat

1999

Journal of Computational Physics

View full text Add to dashboard Cite

The article describes a complete numerical solution of a recently formulated benchmark problem devoted to the parametric study of Rayleigh-Bénard instability in rectangular two-and three-dimensional boxes. The solution is carried out by the spectral Galerkin method with globally defined, three-dimensional, divergent-free basis functions, which satisfy all boundary conditions. The general description of these three-dimensional basis functions, which can be used for a rather wide spectrum of problems, is presented. The results of the parametric calculations are presented as neutral curves showing the dependence of the critical Rayleigh number on the aspect ratio of the cavity. The neutral curves consist of several continuous branches, which belong to different modes of the most dangerous perturbation. The patterns of different perturbations are also reported. The results obtained lead to some new conclusions about the patterns of the most dangerous perturbations and about the similarities between two-and three-dimensional models. Some extensions of the considered benchmark problem are discussed.

show abstract

Section: Discussionmentioning

confidence: 99%

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Gelfgat

1999

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…To apply the integration factor technique to the compact discretization form (10), we multiply (10) by exponential matrix e −At from the left, and e −Bt from the right to obtain (11) Integration of (11) over one time step from t n to t n+1 ≡ t n + Δt, where Δt is the time step, leads to (12) To construct a scheme of rth order truncation error, we approximate the integrand in (12), (13) using a (r − 1)th order Lagrange polynomial at a set of interpolation points t n+1 , t n ,…, t n+2−r : (14) where (15) The specific form of the polynomial (15) at low orders is listed in Table 1. In terms of (τ), (12) takes the form, (16) So the new r-th order implicit schemes are (17) where α 1 , α 0 , α −1 ,···, α −r+2 are coefficients calculated from the integrals of the polynomial in (τ), (18) In Table 2, the value of coefficients, α −j , for schemes of order up to four are listed.…”

Section: Two-dimensionsmentioning

confidence: 99%

“…In terms of (τ), (12) takes the form, (16) So the new r-th order implicit schemes are (17) where α 1 , α 0 , α −1 ,···, α −r+2 are coefficients calculated from the integrals of the polynomial in (τ), (18) In Table 2, the value of coefficients, α −j , for schemes of order up to four are listed.…”

Section: Two-dimensionsmentioning

confidence: 99%

“…If matrices A, C (p) , and B do not commute, one may need to consider the eigenspace of matrices A and B in order to evaluate (22) explicitly [11,12]. Assuming an eigenvalue decomposition (24) is (26) where can be evaluated recursively through integration by parts.…”

Section: Two-dimensionsmentioning

confidence: 99%

“…To obtain the ETD type methods that do not require any commutativity properties on the matrices A and B, and have the same order of operation count as that of the cIIF schemes, one may leave only one of the exponential functions unchanged and apply the polynomial approximation to the rest of the integrand in (12). To illustrate this approach, we first approximate the integrand (27) using a r − 1th order Lagrange polynomial at a set of interpolation points t n+1 , t n ,…, t n+2−r : (28) Then (12) becomes (29) A first order implicit approximation to (τ) of the form (30) leads to a first order implicit scheme (31) A second order implicit approximation to (τ), (32) leads to a second order implicit scheme (33) Like the implicit ETD methods based on the non-compact representation, the nonlinear function of U n+1 in the compact implicit ETD (33) is also multiplied by terms involving the approximated differential operators and their exponentials.…”

Section: Two-dimensionsmentioning

confidence: 99%

See 2 more Smart Citations

Compact integration factor methods in high spatial dimensions

Nie

Wan

Zhang

et al. 2008

Journal of Computational Physics

View full text Add to dashboard Cite

The dominant cost for integration factor (IF) or exponential time differencing (ETD) methods is the repeated vector-matrix multiplications involving exponentials of discretization matrices of differential operators. Although the discretization matrices usually are sparse, their exponentials are not, unless the discretization matrices are diagonal. For example, a two-dimensional system of N × N spatial points, the exponential matrix is of a size of N 2 × N 2 based on direct representations. The vector-matrix multiplication is of O(N 4 ), and the storage of such matrix is usually prohibitive even for a moderate size N. In this paper, we introduce a compact representation of the discretized differential operators for the IF and ETD methods in both two-and three-dimensions. In this approach, the storage and CPU cost are significantly reduced for both IF and ETD methods such that the use of this type of methods becomes possible and attractive for two-or three-dimensional systems. For the case of two-dimensional systems, the required storage and CPU cost are reduced to O(N 2 ) and O(N 3 ), respectively. The improvement on three-dimensional systems is even more significant. We analyze and apply this technique to a class of semi-implicit integration factor method recently developed for stiff reaction-diffusion equations. Direct simulations on test equations along with applications to a morphogen system in two-dimensions and an intra-cellular signaling system in three-dimensions demonstrate an excellent efficiency of the new approach.

show abstract

Stability of convective flows in cavities: solution of benchmark problems by a low‐order finite volume method

Gelfgat

2006

Numerical Methods in Fluids

View full text Add to dashboard Cite

SUMMARYA problem of stability of steady convective flows in rectangular cavities is revisited and studied by a second-order finite volume method. The study is motivated by further applications of the finite volumebased stability solver to more complicated applied problems, which needs an estimate of convergence of critical parameters. It is shown that for low-order methods the quantitatively correct stability results for the problems considered can be obtained only on grids having more than 100 nodes in the shortest direction, and that the results of calculations using uniform grids can be significantly improved by the Richardson's extrapolation. It is shown also that grid stretching can significantly improve the convergence, however sometimes can lead to its slowdown. It is argued that due to the sparseness of the Jacobian matrix and its large dimension it can be effective to combine Arnoldi iteration with direct sparse solvers instead of traditional Krylov-subspace-based iteration techniques. The same replacement in the Newton steady-state solver also yields a robust numerical process, however, it cannot be as effective as modern preconditioned Krylov-subspace-based iterative solvers.

show abstract

Krylov Methods for the Incompressible Navier-Stokes Equations

Cited by 189 publications

References 42 publications

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Compact integration factor methods in high spatial dimensions

Stability of convective flows in cavities: solution of benchmark problems by a low‐order finite volume method

Contact Info

Product

Resources

About