Space-Time Block Preconditioning for Incompressible Flow
Abstract
Parallel-in-time methods have become increasingly popular in the simulation of time-dependent numerical PDEs, allowing for the efficient use of additional message passing interface processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space and time, most notably significantly increased parallelism and faster time to solution (when applicable). However, the development of fast, scalable all-at-once methods has largely been limited to time-dependent (advection-)diffusion problems. This paper introduces the concept of space-time block preconditioning for the all-at-once solution of incompressible flow. By extending well-known concepts of spatial block preconditioning to the space-time setting, we develop a block preconditioner whose application requires the solution of a space-time (advection-)diffusion equation in the velocity block, coupled with a pressure Schur complement approximation consisting of independent spatial solves at each time-step, and a space-time matrix-vector multiplication. The new method is tested on four classical models in incompressible flow. Results indicate perfect scalability in refinement of spatial and temporal mesh spacing, perfect scalability in nonlinear Picard iteration count when applied to a nonlinear Navier--Stokes problem, and minimal overhead in terms of number of preconditioner applications compared with sequential time-stepping.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp, D. Karpeyev, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc, 2019, https://www.mcs.anl.gov/petsc.
2.
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solvers for the generalized Stokes problem, Internat. J. Numer. Methods Fluids, 8 (1988), pp. 869--895, https://doi.org/10.1002/fld.1650080802.
3.
R. Croce, D. Ruprecht, and R. Krause, Parallel-in-space-and-time simulation of the three-dimensional, unsteady Navier-Stokes equations for incompressible flow, in Modeling, Simulation and Optimization of Complex Processes-HPSC 2012, Springer, Cham, 2014, pp. 13--23, https://doi.org/10.1007/978-3-319-09063-4_2.
4.
F. Danieli, STBP Incompressible Flow, 2020, https://gitlab.com/fdanieli/stbp-incompressible-flow.
5.
H. Elman, D. Silvester, and A. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Numer. Math. Sci. Comput., Oxford University Press, Oxford, 2014, https://doi.org/10.1093/acprof:oso/9780199678792.001.0001.
6.
H. Elman and R. Tuminaro, Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations, Electron. Trans. Numer. Anal., 35 (2009), pp. 257--280.
7.
H. C. Elman, A. Ramage, and D. J. Silvester, IFISS: A computational laboratory for investigating incompressible flow problems, SIAM Rev., 56 (2014), pp. 261--273, https://doi.org/10.1137/120891393.
8.
R. D. Falgout, S. Friedhoff, T. V. Kolev, S. P. MacLachlan, and J. B. Schroder, Parallel time integration with multigrid, SIAM J. Sci. Comput., 36 (2014), pp. C635--C661, https://doi.org/10.1137/130944230.
9.
R. D. Falgout, S. Friedhoff, T. V. Kolev, S. P. MacLachlan, J. B. Schroder, and S. Vandewalle, Multigrid methods with space-time concurrency, Comput. Vis. Sci., 18 (2017), pp. 123--143, https://doi.org/10.1007/s00791-017-0283-9.
10.
P. Fischer, F. Hecht, and Y. Maday, A parareal in time semi-implicit approximation of the Navier-Stokes equations, in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 40, Springer, Cham, 2005, pp. 433--440, https://doi.org/10.1007/3-540-26825-1_44.
11.
M. J. Gander, 50 years of time parallel time integration, in Multiple Shooting and Time Domain Decomposition Methods, T. Carraro, M. Geiger, S. Körkel, and R. Rannacher, eds., Springer, Cham, 2015, pp. 69--113, https://doi.org/10.1007/978-3-319-23321-5_3.
12.
M. J. Gander and L. Halpern, Time parallelization for nonlinear problems based on diagonalization, in Domain Decomposition Methods in Science and Engineering XXIII, C.-O. Lee, X.-C. Cai, D. E. Keyes, H. H. Kim, A. Klawonn, E.-J. Park, and O. B. Widlund, eds., Springer, Cham, 2017, pp. 163--170.
13.
M. J. Gander, L. Halpern, J. Ryan, and T. T. B. Tran, A direct solver for time parallelization, in Domain Decomposition Methods in Science and Engineering XXII, T. Dickopf, M. J. Gander, L. Halpern, R. Krause, and L. F. Pavarino, eds., Springer, Cham, 2016, pp. 491--499.
14.
M. J. Gander and S. Vandewalle, Analysis of the parareal time-parallel time-integration method, SIAM J. Sci. Comput., 29 (2007), pp. 556--578, https://doi.org/10.1137/05064607X.
15.
A. Goddard and A. Wathen, A note on parallel preconditioning for all-at-once evolutionary PDEs, Electron. Trans. Numer. Anal., 51 (2019), pp. 135--150.
16.
A. Greenbaum, V. Pták, and Z. Strakoš, Any nonincreasing convergence curve is possible for GMRES, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 465--469, https://doi.org/10.1137/S0895479894275030.
17.
G. Horton and S. Vandewalle, A space-time multigrid method for parabolic partial differential equations, SIAM J. Sci. Comput., 16 (1995), pp. 848--864, https://doi.org/10.1137/0916050.
18.
HYPRE: Scalable Linear Solvers and Multigrid Methods, www.llnl.gov/casc/hypre/.
19.
IFISS: Incompressible Flow and Iterative Solver Software, https://personalpages.manchester.ac.uk/staff/david.silvester/ifiss/default.htm.
20.
D. Kay, D. Loghin, and A. Wathen, A preconditioner for the steady-state Navier--Stokes equations, SIAM J. Sci. Comput., 24 (2002), pp. 237--256, https://doi.org/10.1137/S106482759935808X.
21.
J.-L. Lions, Y. Maday, and G. Turinici, Résolution d'EDP par un schéma en temps $\langle\langle$pararéel$\rangle\rangle$, C. R. Math. Acad. Sci. Paris, 332 (2001), pp. 661--668, https://doi.org/10.1016/S0764-4442(00)01793-6.
22.
J. Liu and S.-L. Wu, A fast block $\alpha$-circulant preconditioner for all-at-once system from wave equations, SIAM J. Matrix Anal. Appl., 41 (2020), pp. 1912--1943, https://doi.org/10.1137/19M1309869.
23.
T. A. Manteuffel, S. Münzenmaier, J. Ruge, and B. Southworth, Nonsymmetric reduction-based algebraic multigrid, SIAM J. Sci. Comput., 41 (2019), pp. S242--S268, https://doi.org/10.1137/18M1193761.
24.
T. A. Manteuffel, J. Ruge, and B. S. Southworth, Nonsymmetric algebraic multigrid based on local approximate ideal restriction ($\ell$AIR), SIAM J. Sci. Comput., 40 (2018), pp. A4105--A4130, https://doi.org/10.1137/17M1144350.
25.
E. McDonald, J. Pestana, and A. Wathen, Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations, SIAM J. Sci. Comput., 40 (2018), pp. A1012--A1033, https://doi.org/10.1137/16M1062016.
26.
MFEM: Modular Finite Element Methods, https://doi.org/10.11578/dc.20171025.1248.
27.
Z. Miao, Y.-L. Jiang, and Y.-B. Yang, Convergence analysis of a parareal-in-time algorithm for the incompressible non-isothermal flows, Int. J. Comput. Math., 96 (2019), pp. 1398--1415, https://doi.org/10.1080/00207160.2018.1498484.
28.
C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617--629, https://doi.org/10.1137/0712047.
29.
É. Picard, Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires, J. Math. Pures Appl., 9 (1893), pp. 217--272.
30.
A. Quarteroni, Numerical Models for Differential Problems, Springer, Cham, 2017, https://doi.org/10.1007/978-3-319-49316-9.
31.
S. Rhebergen and B. Cockburn, A space--time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys., 231 (2012), pp. 4185--4204, https://doi.org/10.1016/j.jcp.2012.02.011.
32.
S. Rhebergen, B. Cockburn, and J. J. Van Der Vegt, A space--time discontinuous Galerkin method for the incompressible Navier--Stokes equations, J. Comput. Phys., 233 (2013), pp. 339--358, https://doi.org/10.1016/j.jcp.2012.08.052.
33.
J. W. Ruge and K. Stüben, Algebraic Multigrid, SIAM, Philadelphia, 1987, pp. 73--130, https://doi.org/10.1137/1.9781611971057.ch4.
34.
Y. Saad, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461--469, https://doi.org/10.1137/0914028.
35.
Y. Saad and M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869, https://doi.org/10.1137/0907058.
36.
D. Silvester, H. Elman, D. Kay, and A. Wathen, Efficient preconditioning of the linearized Navier-Stokes equations for incompressible flow, J. Comput. Appl. Math., 128 (2001), pp. 261--279, https://doi.org/10.1016/S0377-0427(00)00515-X.
37.
A. A. Sivas, B. S. Southworth, and S. Rhebergen, AIR algebraic multigrid for a space-time hybridizable discontinuous Galerkin discretization of advection (-diffusion), SIAM J. Sci. Comput., 43 (2021), pp. A3393--A3416.
38.
B. S. Southworth, A. A. Sivas, and S. Rhebergen, On fixed-point, Krylov, and 2 x 2 block preconditioners for nonsymmetric problems, SIAM J. Matrix Anal. Appl., 41 (2020), pp. 871--900, https://doi.org/10.1137/19M1298317.
39.
M. Stoll and A. Wathen, All-at-once solution of time-dependent Stokes control, J. Comput. Phys., 232 (2013), https://doi.org/10.1016/j.jcp.2012.08.039.
40.
J. Trindade and J. Pereira, Parallel-in-time simulation of the unsteady Navier--Stokes equations for incompressible flow, Internat. J. Numer. Methods Fluids, 45 (2004), pp. 1123--1136, https://doi.org/10.1002/fld.732.
41.
J. Trindade and J. Pereira, Parallel-in-time simulation of two-dimensional, unsteady, incompressible laminar flows, Numer. Heat Transf. B Fund., 50 (2006), pp. 25--40, https://doi.org/10.1080/10407790500459379.
42.
Q. Wang, S. A. Gomez, P. J. Blonigan, A. L. Gregory, and E. Y. Qian, Towards scalable parallel-in-time turbulent flow simulations, Phys. Fluids, 25 (2013), 110818, https://doi.org/10.1063/1.4819390.
43.
A. Wathen and T. Rees, Chebyshev semi-iteration in preconditioning for problems including the mass matrix, Electron. Trans. Numer. Anal., 34 (2008-2009), pp. 125--135.
44.
A. J. Wathen, Realistic eigenvalue bounds for the Galerkin mass matrix, IMA J. Numer. Anal., 7 (1987), pp. 449--457, https://doi.org/10.1093/imanum/7.4.449.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A337 - A363
DOI: 10.1137/21M1390773
ISSN (online): 1095-7197
Copyright
© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 11 January 2021
Accepted: 11 October 2021
Published online: 8 February 2022
Keywords
MSC codes
Authors
Funding Information
Los Alamos National Laboratory https://doi.org/10.13039/100008902
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/L015803/1
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- On Properties of Adjoint Systems for Evolutionary PDEsJournal of Nonlinear Science, Vol. 34, No. 5 | 16 August 2024
- A novel α-absolute value preconditioner for all-at-once systems from heat equationsComputers & Mathematics with Applications, Vol. 165 | 1 Jul 2024
- Augmented Lagrangian Acceleration of Global-in-Time Pressure Schur Complement Solvers for Incompressible Oseen EquationsJournal of Mathematical Fluid Mechanics, Vol. 26, No. 2 | 28 March 2024
- PinT Preconditioner for Forward-Backward Evolutionary EquationsSIAM Journal on Matrix Analysis and Applications, Vol. 44, No. 4 | 30 November 2023
- On the Design of Global-in-Time Newton-Multigrid-Pressure Schur Complement Solvers for Incompressible Flow ProblemsJournal of Mathematical Fluid Mechanics, Vol. 25, No. 3 | 26 June 2023
- Parallel-in-time preconditioner for the Sinc-Nyström systemsSIAM Journal on Scientific Computing, Vol. 44, No. 4 | 15 August 2022
- A well-conditioned direct PinT algorithm for first- and second-order evolutionary equationsAdvances in Computational Mathematics, Vol. 48, No. 3 | 6 April 2022
View Options
Get Access
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].