Pith. sign in

REVIEW 3 major objections 4 minor 52 references

An Augmented Lagrangian Preconditioner for Navier--Stokes Equations with Runge--Kutta in Time

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper builds an augmented Lagrangian block-triangular preconditioner that keeps flexible GMRES iteration counts bounded as viscosity, mesh size, time step, and Runge-Kutta stage count vary, with linear problem-size scaling under an…

desk verdict The paper is a solid, honest numerical methods contribution with wide experiments, but the central robustness claim rests on a heuristic Schur approximation that is only tested down to nu=1/2500 and a time-step dependence that is never varied independently. read the letter →

arxiv 2506.04451 v1 pith:NMN2FY3A submitted 2025-06-04 math.NA cs.NA

Santolo Leveque , Yunhui He , Maxim Olshanskii This is my paper
classification math.NAcs.NA MSC 65F0865F1065N2265L06
keywords Navier-StokesequationsRunge-KuttatimeintegrationaugmentedLagrangianpreconditionersaddle-pointsystemspreconditioningflexibleGMRESmultigridfiniteelementmethods
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the many large linear systems produced when an implicit Runge-Kutta method time-steps the incompressible Navier-Stokes equations can all be solved by one robust preconditioner. Each time step couples all Runge-Kutta stages together in a generalized saddle-point system, and each Newton linearization produces another such system. The authors construct an augmented Lagrangian block-triangular preconditioner, treat the augmented momentum block with a block Gauss-Seidel method, and approximate the Schur complement with a Stokes-inspired formula. Their numerical experiments show iteration counts that stay essentially fixed as viscosity, mesh size, time step, and stage count vary, and CPU time that grows linearly with problem size once the inner blocks are handled by one multigrid cycle. If the claim holds, high-order A-stable Runge-Kutta time integration of incompressible flows becomes practical at large scale without redesigning the solver for each flow regime.

What carries the argument

The central object is the generalized saddle-point stage system (3.8), whose (1,1)-block couples all Runge-Kutta stages through the matrices Phi and whose off-diagonal blocks carry the incompressibility constraint through $\Delta$ t A_RK tensor B. The preconditioner augments the (1,1)-block by gamma Psi_1 $W^{{-1}}$ Psi_2 with W = $\Delta$ t A_RK tensor W, then applies the Sherman-Morrison-Woodbury identity to rewrite the inverse Schur complement as gamma $W^{{-1}}$ + (Psi_2 $Phi^{{-1}}$ Psi_1)^{-1}. The inner inverse is approximated by equation (4.7), the heuristic Stokes-derived formula I_s tensor $K_p^{{-1}}$ + nu $\Delta$ t A_RK tensor $M_p^{{-1}}$. The outer iteration is flexible GMRES, and the augmented momentum block is approximated by block Gauss-Seidel, with each diagonal block solved either exactly or by one multigrid cycle.

What would settle it

Run the lid-driven cavity or cylinder benchmark with the same inexact multigrid setup at nu = $10^{{-5}}$, refining the mesh and recording average FGMRES iterations; if the iteration count doubles as h halves or grows steadily as nu decreases, instead of staying bounded near the 10-30 range reported here, the central robustness claim is refuted.

Watch

Extended reading notes

Core claim

The paper claims that for each Newton linearization of the Runge-Kutta stage system, the augmented Lagrangian preconditioner with augmented (1,1)-block and the Schur complement approximation from equations (4.7) and (4.8) makes the preconditioned generalized saddle-point system robust with respect to both problem parameters and the choice of Runge-Kutta method. In the reported tests, average flexible GMRES iterations stay below 33 with exact diagonal solves and at most 31 with one multigrid cycle per augmented block, for Gauss, Lobatto IIIC, and Radau IIA methods with s = 2, 3, 4 stages and viscosities down to 1/2500. Newton's method converges in roughly 1 to 2 iterations per time step in the inexact-multigrid experiments, and the measured CPU times scale linearly with the number of degrees of freedom.

Load-bearing premise

The load-bearing premise is that the Stokes-derived Schur complement formula (4.7) remains accurate enough for convection-dominated Navier-Stokes stage systems; if that approximation degrades at low viscosity or high Reynolds number, iteration counts would grow.

Editorial extensions

If this is right

  • Implicit Runge-Kutta stage systems for Navier-Stokes can be solved by one preconditioner whose iteration counts stay bounded as viscosity, mesh size, time step, and stage count vary.
  • With an inexact multigrid application for the augmented momentum blocks, total CPU time scales approximately linearly with the number of degrees of freedom.
  • The method works across Gauss, Lobatto IIIC, and Radau IIA families, so choosing a high-order, A-stable Runge-Kutta method does not require redesigning the linear solver.
  • Increasing the augmentation parameter gamma reduces outer iterations, while Newton's method still converges in a small number of iterations per time step.
  • The algebraic form of the augmentation coincides with a grad-div stabilized discretization, so the same preconditioner applies to those formulations as well.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The heuristic Schur complement formula (4.7) is the piece most likely to limit the method at extreme Reynolds numbers; a field-of-values analysis for convection-dominated stage systems could turn the numerical robustness into a provable bound, but the paper does not supply it.
  • Because the stage coupling enters primarily through Delta t A_RK in the off-diagonal blocks, the same preconditioner structure should transfer to 3D flows and to stage-parallel implementations once the augmented momentum blocks are handled by a scalable smoother; neither direction is tested here.
  • The observed decrease in pressure error with larger gamma at fixed mesh suggests gamma could be tuned dynamically to trade linear iterations against pressure accuracy, a strategy the paper leaves unexplored.
  • If embedded in a general Runge-Kutta finite element framework, this solver would give practitioners high-order A-stable time stepping for incompressible flow at near-linear cost, removing one practical obstacle to replacing second-order multistep integrators.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops an augmented Lagrangian block-triangular preconditioner for the large sparse generalized saddle-point systems that arise when an implicit Runge--Kutta method is used for time discretization of the incompressible Navier--Stokes equations and Newton's method is applied to the stage equations. The preconditioner uses an augmented (1,1)-block, a block Gauss--Seidel approximation of that block with either exact or multigrid inner solves, and a Schur complement approximation based on pressure stiffness and mass operators. Numerical experiments with Radau IIA, Gauss, and Lobatto IIIC methods on a lid-driven cavity and on flow around a cylinder report FGMRES iteration counts in a narrow band and, for the multigrid version, CPU times that scale roughly linearly with problem size.

Significance. If the robustness claims hold, this is a practically valuable contribution: fully implicit Runge--Kutta time stepping for incompressible flow currently has few linear solver options beyond monolithic multigrid, and the paper demonstrates a credible algebraic alternative for a wide range of viscosities, mesh sizes, stage counts, and Runge--Kutta families. The algebraic equivalence of the augmented system (4.4) to the original saddle-point system (3.8) is transparent and correct, and the numerical study is extensive, including a realistic benchmark problem and use of the public Firedrake/Irksome packages. The main weakness is that the central robustness claim rests on a Schur complement approximation that the paper itself labels heuristic and that was designed for the Stokes limit; the evidence presented does not yet support the unqualified parameter-robustness statement made in the introduction.

major comments (3)
  1. [Section 4, Eq. (4.7)] The approximation eS_int^{-1} = Is ⊗ K_p^{-1} + νΔt A_RK ⊗ M_p^{-1} is carried over from the Stokes setting of [29], and the paper explicitly calls it heuristic. The exact object S_int contains the full linearized convection--diffusion operator through Φ, so the quality of the replacement of the convection contribution by a pressure-Laplacian term is exactly what must be validated in the convection-dominated regime. Tables 5.3--5.10 only go down to ν = 1/2500, and no experiment isolates the effect of Δt at fixed h. The introduction's unqualified claim that the solver is "robust with respect to both the problem parameters and the choice of Runge--Kutta method" therefore goes beyond the evidence. Please provide either a quantitative study of (4.7) as ν → 0 and as Δt varies, or restrict the robustness claim to the tested parameter range.
  2. [Section 5.1] The paper introduces Local Projection Stabilization and defines the stabilization matrix Qu, but it never states explicitly how Qu enters the algebraic system (3.7)--(3.8) or whether the preconditioner is applied to the stabilized or unstabilized operator. If Qu is added to the (1,1) block, the derivation of Section 4 does not directly apply to the system actually solved; if Qu is not added, the high-Reynolds tests are not stabilized despite the stated intent. In either case, the numerical results in Tables 5.3--5.6 are not fully auditable. Please specify the stabilized discrete system, the corresponding modified Φ, and any changes to the approximations in (4.6)--(4.8), or remove LPS from the algebraic formulation.
  3. [Section 5.2.1] The claimed robustness with respect to the time step is not directly demonstrated. In Section 5.2.1 the number of time steps is fixed at nt = 50 in Tables 5.7--5.9, so only h varies while Δt is fixed. In Section 5.1.2, nt is chosen through the condition Δt ≤ h^{qFE}/qRK, so h and Δt change simultaneously. No experiment varies Δt at fixed h. Since the Schur complement approximation (4.7) depends explicitly on Δt, an experiment that isolates the dependence on Δt is needed to support the time-step robustness claim.
minor comments (4)
  1. [Section 4, Eq. (4.2)] The statement that for the ideal block-triangular preconditioner P = [[Φ, Ψ1], [0, −S]] "the preconditioned matrix has all eigenvalues equal to 1" is not correct. For the simple 2x2 case with Φ = 1, Ψ1 = 1, Ψ2 = 1, and Θ = 0, one obtains S = 1 and P^{-1}A has eigenvalues 1 ± √2. This does not invalidate the numerical method, but the claim should be corrected or replaced by the standard field-of-values or minimal-polynomial statement for block-triangular preconditioners.
  2. [Acknowledgements] The word "greatfully" should be "gratefully".
  3. [Table 5.10] The table header contains the typo "freedon"; it should be "freedom".
  4. [Section 3, Eq. (3.7)] The role of Hu in the Newton linearization would be clearer if the text noted explicitly that Hu is the discrete derivative of the convection term with respect to the velocity unknown, rather than describing it only as "second-order information".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the robustness and linear-scaling claims are empirical, and the imported Schur-complement heuristic is transparently labeled and independently testable.

full rationale

The paper's central claim is that the augmented Lagrangian preconditioner is robust in viscosity, mesh size, time step, and Runge-Kutta stage count, and that the resulting solver scales linearly. This claim is supported by measured iteration counts and CPU times in Section 5, not derived by construction from the cited prior work. No parameter is fitted to force the iteration counts: gamma is selected by hand (γ = 1, 10, 100) and then held fixed while h, ν, s, and the Runge-Kutta family are varied. The key input that could raise circularity concerns is the heuristic Schur approximation in Eq. (4.7), attributed to the first author's earlier work [29] for the Stokes equations in the very viscous limit. The paper explicitly calls this approximation heuristic, states its provenance, and then tests its behavior in the Navier-Stokes setting over a range of parameters. This is an extrapolation with transparent assumptions, not a self-citation used to forbid alternatives or to define the target result into existence. The exact Schur complement and its approximation are not equal by construction; the approximation contains pressure stiffness and pressure mass terms, while the exact object contains the full Oseen operator through Φ, so the robustness observed in Tables 5.3-5.10 is a genuine numerical outcome. The limited viscosity range (down to ν = 1/2500) and the heuristic status of (4.7) are correctness risks or limitations, but they are not circularity. The paper also reproduces benchmark problems (lid-driven cavity, channel around a cylinder), giving external falsifiability for the empirical claims. No equation is self-definitional, no fitted quantity is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. The manuscript is self-contained as a numerical study of a preconditioner assembled from published components, and the load-bearing contribution is the new stage-system preconditioner with its numerical validation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The core algebraic manipulation (augmentation and block triangular preconditioner) is standard and easily verified; the two genuinely imported ingredients are the heuristic Schur complement formula (4.7) and the block Gauss-Seidel/multigrid approximation of the (1,1)-block. The only hand-tuned numerical parameter is the augmentation gamma, set to 1 for most runs. No invented physical or mathematical entities are introduced.

free parameters (1)
  • Augmentation parameter gamma = 1 (default), 10, 100 tested
    The augmented Lagrangian parameter enters the preconditioner (4.6) and the augmented system (4.4). The paper chooses gamma=1 for most tests without a theory-based selection rule; robustness is shown empirically across gamma values, but no analysis fixes its optimal value.
assumptions (4)
  • standard math The Runge-Kutta coefficient matrix A_RK is invertible.
    Assumed in Section 2 to allow W and S approximations using A_RK^{-1}; the paper notes this is common and not restrictive.
  • domain assumption The finite element pair is inf-sup stable.
    Assumed in Section 3; tests use Taylor-Hood Q2-Q1 and Scott-Vogelius P4-P3 elements.
  • ad hoc to paper The heuristic Schur approximation eS_int^{-1} = Is ⊗ K_p^{-1} + ν∆t A_RK ⊗ M_p^{-1} approximates S_int^{-1}.
    Adopted from [29] for Stokes in the very viscous limit; used without derivation or analysis for Navier-Stokes in Eq (4.7). This is load-bearing for the preconditioner's Schur block.
  • ad hoc to paper Block Gauss-Seidel with multigrid approximates the augmented (1,1)-block adequately.
    Section 4 and 5; no convergence theory is provided for the inexact block solver applied to the convection-dominated augmented momentum block.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An Augmented Lagrangian Preconditioner for Navier--Stokes Equations with Runge--Kutta in Time." pith.science (2026). https://pith.science/paper/NMN2FY3A

@misc{pith2026250604451,
  author       = {Pith},
  title        = {Pith review of: An Augmented Lagrangian Preconditioner for Navier--Stokes Equations with Runge--Kutta in Time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMN2FY3A}},
  note         = {Machine review of arXiv:2506.04451}
}
read the original abstract

We consider a Runge--Kutta method for the numerical time integration of the nonstationary incompressible Navier--Stokes equations. This yields a sequence of nonlinear problems to be solved for the stages of the Runge--Kutta method. The resulting nonlinear system of differential equations is discretized using a finite element method. To compute a numerical approximation of the stages at each time step, we employ Newton's method, which requires the solution of a large and sparse generalized saddle-point problem at each nonlinear iteration. We devise an augmented Lagrangian preconditioner within the flexible GMRES method for solving the Newton systems at each time step. The preconditioner can be applied inexactly with the help of a multigrid routine. We present numerical evidence of the robustness and efficiency of the proposed strategy for different values of the viscosity, mesh size, time step, and number of stages of the Runge--Kutta method.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [29]

    Leveque, L

    S. Leveque, L. Bergamaschi, ´A. Mart ´ ınez, and J. W. Pearson,Parallel-in-Time Solver for the All-at-Once Runge–Kutta Discretization, SIAM J. Matrix Anal. Appl., 45, pp. 1902–1928, 2024

    work page 1902

  2. [1]

    Abu-Labdeh, S

    R. Abu-Labdeh, S. MacLachlan, and P. E. Farrell, Monolithic multigrid for implicit Runge– Kutta discretizations of incompressible fluid flow , J. Comput. Phys., 478, p. 111961, 2023

    work page 2023

  3. [2]

    Axelsson, I

    O. Axelsson, I. Dravins, and M. Neytcheva, Stage-parallel preconditioners for implicit Runge– Kutta methods of arbritary high order. Linear problems , Uppsala University, Technical Report from the Department of Information Technology 2022-004, 2022

    work page 2022

  4. [3]

    Becker and M

    R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equa- tions based on local projections , Calcolo, 38, pp. 173–199, 2001

    work page 2001

  5. [4]

    Becker and B

    R. Becker and B. Vexler, Optimal control of the convection–diffusion equation using stabilized finite element methods , Numer. Math., 106, pp. 349–367, 2007

    work page 2007

  6. [5]

    Benzi, G

    M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems , Acta Numerica, 14, pp. 1–137, 2005

    work page 2005

  7. [6]

    Benzi and M

    M. Benzi and M. A. Olshanskii, An augmented Lagrangian-based approach to the Oseen prob- lem, SIAM J. Sci. Comput., 28, pp. 2095–2113, 2006

    work page 2006

  8. [7]

    Benzi and M

    M. Benzi and M. A. Olshanskii, Field-of-values convergence analysis of augmented Lagrangian preconditioners for the linearized Navier–Stokes problem , SIAM J. Numer. Anal., 49, pp. 770–788, 2011

    work page 2011

Show all 52 references
  1. [8]

    Boonen, J

    T. Boonen, J. Van lent, and S. Vandewalle, An algebraic multigrid method for high order time-discretizations of the div-grad and the curl-curl equations , Appl. Numer. Math., 59, pp. 507–521, 2009

    2009

  2. [9]

    Boyle, M

    J. Boyle, M. Mihajlovi´ c, and J. Scott, HSL MI20: an efficient AMG preconditioner for finite element problems in 3D , Int. J. Numer. Meth. Eng., 82, pp. 64–98, 2010

    2010

  3. [10]

    Braack and E

    M. Braack and E. Burman, Local projection stabilization for the Oseen problem and its inter- pretation as a variational multiscale method , SIAM J. Numer. Anal., 43, pp. 2544–2566, 2006

    2006

  4. [11]

    A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov–Galerkin formulations for con- vection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Eng., 32, pp. 199–259, 1982

    1982

  5. [12]

    J. C. Butcher, Numerical methods for ordinary differential equations , 3rd edition, John Wiley & Sons, Ltd, 2016

    2016

  6. [13]

    H. C. Elman, D. J. Silvester, and A. J. Wathen: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2nd edition, Oxford University Press, 2014

    2014

  7. [14]

    P. E. Farrell, R. C. Kirby, and J. Marchena-Men´ endez, Irksome: automating Runge–Kutta time-stepping for finite element methods , ACM Trans. Math. Softw., 47, Art. 30, 2021

    2021

  8. [15]

    P. E. Farrell, L. Mitchell, and F. Wechsung, An augmented Lagrangian preconditioner for the 3D stationary incompressible Navier-Stokes equations at high Reynolds numbers , SIAM J. Sci. Comput., 41, pp. A3075–A3096, 2019

    2019

  9. [16]

    L. P. Franca and S. L. Frey, Stabilized finite element methods: II. The incompressible Navier– Stokes equations, Comput. Methods Appl. Mech. Eng., 99, pp. 209–233, 1992

    1992

  10. [17]

    G. H. Golub and R. S. Varga, Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods, Part I , Numer. Math., 3, pp. 147–156, 1961

    1961

  11. [18]

    G. H. Golub and R. S. Varga, Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods, Part II, Numer. Math., 3, pp. 157–168, 1961. 18 S. LEVEQUE, Y. HE, AND M. OLSHANSKII

    1961

  12. [19]

    J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195, pp. 6011–6045, 2006

    2006

  13. [20]

    Hairer and G

    E. Hairer and G. Wanner, Solving ordinary differential equations II: stiff and differential- algebraic problems, 2nd edition, Springer Berlin, Heidelberg, 1996

    1996

  14. [21]

    Huang and J

    F. Huang and J. Shen, Stability and error analysis of a class of high-order IMEX schemes for Navier–Stokes equations with periodic boundary conditions , SIAM J. Numer. Anal., 59, pp. 2926–2954, 2021

    2021

  15. [22]

    I. C. F. Ipsen, A note on preconditioning nonsymmetric matrices , SIAM J. Sci. Comput., 23, pp. 1050–1051, 2001

    2001

  16. [23]

    Johnson and J

    C. Johnson and J. Saranen, Streamline diffusion methods for the incompressible Euler and Navier–Stokes equations, Math. Comp., 47, pp. 1–18, 1986

    1986

  17. [24]

    R. C. Kirby, On the convergence of monolithic multigrid for implicit Runge–Kutta time step- ping of finite element problems , SIAM J. Sci. Comput., 46, pp. S22–S45, 2024

    2024

  18. [25]

    R. C. Kirby and S. P. MacLachlan: Extending Irksome: improvements in automated Runge– Kutta time stepping for finite element methods , https://arxiv.org/abs/2403.08084, 2024

    2024 arXiv

  19. [26]

    Kressner, S

    D. Kressner, S. Massei, and J. Zhu, Improved parallel-in-time integration via low-rank updates and interpolation, arXiv:2204.03073, 2023

    2023 arXiv

  20. [27]

    J. D. Lambert, Numerical method for ordinary differential systems: the initial value problem , John Wiley & Sons, Ltd, New York, 1991

    1991

  21. [28]

    Leveque, M

    S. Leveque, M. Benzi, and P. E. Farrell: An Augmented Lagrangian Preconditioner for the Control of the Navier–Stokes Equations , arXiv preprint arXiv:2408.05095, 2024

    2024 arXiv

  22. [30]

    Mardal, T

    K.-A. Mardal, T. K. Nilssen, and G. A. Staff, Order-optimal preconditioners for implicit Runge–Kutta schemes applied to parabolic PDEs, SIAM J. Sci. Comput., 29, pp. 361–375, 2007

    2007

  23. [31]

    Marion and T

    M. Marion and T. Roger, Navier–Stokes equations: theory and approximation , Handbook of numerical analysis, 6, pp. 503–689, 1998

    1998

  24. [32]

    Matthies and L

    G. Matthies and L. Tobiska, Local projection type stabilization applied to inf–sup stable dis- cretizations of the Oseen problem , IMA J. Numer. Anal., 35, pp. 239–269, 2015

    2015

  25. [33]

    Munch, I

    P. Munch, I. Dravins, M. Kronbichler, and M. Neytcheva, Stage-parallel fully implicit Runge– Kutta implementations with optimal multilevel preconditioners at the scaling limit , SIAM J. Sci. Comput., pp. S71–S96, 2022

    2022

  26. [34]

    M. F. Murphy, G. H. Golub, and A. J. Wathen, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21, pp. 1969–1972, 2000

    1969

  27. [35]

    M. A. Olshanskii, A low order Galerkin finite element method for the Navier–Stokes equa- tions of steady incompressible flow: a stabilization issue and iterative methods , Computer Methods in Applied Mechanics and Engineering, 191, pp. 5515–5536, 2002

    2002

  28. [36]

    Md. M. Rana, V. E. Howle, K. Long, A. Meek, and W. Milestone, A new block preconditioner for implicit Runge–Kutta methods for parabolic PDE problems , SIAM J. Sci. Comput., 43, pp. S475–S495, 2021

    2021

  29. [37]

    Rathgeber, D

    F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. Mcrae, G.-T. Bercea, G. R. Markall, and P. H. J. Kelly, Firedrake: automating the finite element method by composing abstractions, ACM Trans. Math. Softw., 43, Art. 24, 2016

    2016

  30. [38]

    Rosseel, T

    E. Rosseel, T. Boonen, and S. Vandewalle, Algebraic multigrid for stationary and time- dependent partial differential equations with stochastic coefficients , Numer. Linear Alg. Appl., 15, pp. 141–163, 2008

    2008

  31. [39]

    Saad, A flexible inner–outer preconditioned GMRES algorithm , SIAM J

    Y. Saad, A flexible inner–outer preconditioned GMRES algorithm , SIAM J. Sci. Comput., 14, pp. 461–469, 1993

    1993

  32. [40]

    Saad and M

    Y. Saad and M. H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , SIAM J. Sci. Stat. Comput., 7, pp. 856–869, 1986

    1986

  33. [41]

    Sch¨ afer, S

    M. Sch¨ afer, S. Turek, F. Durst, E. Krause, and R. Rannacher, Benchmark Computations of Laminar Flow Around a Cylinder . In: E. H. Hirschel (eds.), Flow Simulation with High- Performance Computers II: DFG Priority Research Programme Results 1993–1995 , 48, pp. 547–566, Vieweg...

    1993

  34. [42]

    L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials , ESAIM: Math. Mod. Num. Anal., 19, pp. 111–143, 1985

    1985

  35. [43]

    Silvester and A

    D. Silvester and A. Wathen, Fast iterative solution of stabilised Stokes systems, Part II: using general block preconditioners, SIAM J. Numer. Anal., 31, pp. 1352–1367, 1994

    1994

  36. [44]

    B. S. Southworth, O. Krzysik, and W. Pazner: Fast Solution of Fully Implicit Runge–Kutta AL PRECONDITIONER FOR NA VIER–STOKES RUNGE–KUTTA IN TIME 19 and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs, SIAM J. Sci. Comput., 44, pp. A636–A663, 2022

    2022

  37. [45]

    B. S. Southworth, O. Krzysik, W. Pazner, and H. De Sterck, Fast solution of fully implicit Runge–Kutta and discontinuous Galerkin in time for numerical PDEs, Part I: the linear setting, SIAM J. Sci. Comput., 44, pp. A416–A443, 2022

    2022

  38. [46]

    G. A. Staff, K.-A. Mardal, and T. K. Nilssen, Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs , Model. Identif. Control, 27, pp. 109–123, 2006

    2006

  39. [47]

    Temam, Navier–Stokes equations: theory and numerical analysis , AMS Chelsea Publishing, 1988

    R. Temam, Navier–Stokes equations: theory and numerical analysis , AMS Chelsea Publishing, 1988

    1988

  40. [48]

    Tobiska and G

    L. Tobiska and G. Lube, A modified streamline diffusion method for solving the stationary Navier–Stokes equations, Numer. Math., 59, pp. 13–29, 1991

    1991

  41. [49]

    Van Lent and S

    J. Van Lent and S. Vandewalle, Multigrid methods for implicit Runge–Kutta and boundary value method discretizations of parabolic PDEs , SIAM J. Sci. Comput., 27, pp. 67–92, 2005

    2005

  42. [50]

    Wathen and T

    A. Wathen and T. Rees, Chebyshev semi-iteration in preconditioning for problems including the mass matrix , Electron. Trans. Numer. Anal., 34, pp. 125–135, 2009

    2009

  43. [51]

    Wathen and D

    A. Wathen and D. Silvester, Fast iterative solution of stabilised Stokes systems, Part I: using simple diagonal preconditioners, SIAM J. Numer. Anal., 30, pp. 630–649, 1993

    1993

  44. [52]

    William, Introduction to the numerical analysis of incompressible viscous flows , SIAM, 2008

    L. William, Introduction to the numerical analysis of incompressible viscous flows , SIAM, 2008

    2008

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.