Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs

“Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs”

Authors: Gunnar A. Staff, Kent-Andre Mardal and Trygve K. Nilssen,
Affiliation: Simula Research Laboratory
Reference: 2006, Vol 27, No 2, pp. 109-123.

Keywords: Runge-Kutta methods, PDEs, preconditioning, order-optimal methods

Abstract: Recently, the authors introduced a preconditioner for the linear systems that arise from fully implicit Runge-Kutta time stepping schemes applied to parabolic PDEs (9). The preconditioner was a block Jacobi preconditioner, where each of the blocks were based on standard preconditioners for low-order time discretizations like implicit Euler or Crank-Nicolson. It was proven that the preconditioner is optimal with respect to the timestep and the discretization parameter in space. In this paper we will improve the convergence by considering other preconditioners like the upper and the lower block Gauss-Seidel preconditioners, both in a left and right preconditioning setting. Finally, we improve the condition number by using a generalized Gauss-Seidel preconditioner.

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PDF (1317 Kb) DOI: 10.4173/mic.2006.2.3

DOI forward links to this article:

[1] MAGALI RIBOT and MICHELLE SCHATZMAN (2011), doi:10.1142/S1793744211000436

[2] Kent-Andre Mardal and Ragnar Winther (2011), doi:10.1002/nla.716

[3] K. A. Mardal, T. K. Nilssen and G. A. Staff (2007), doi:10.1137/05064093X

[4] Trygve K. Nilssen, Gunnar A. Staff and Kent-Andre Mardal (2011), doi:10.1002/num.20582

[5] Hao Chen (2014), doi:10.1007/s10543-014-0467-3

[6] Hao Chen (2014), doi:10.1002/nla.1960

[7] Owe Axelsson, Radim Blaheta and Roman Kohut (2015), doi:10.1002/nla.2015

[8] Hao Chen (2015), doi:10.1016/j.apm.2015.11.037

[9] Steffen Basting and Eberhard Bänsch (2017), doi:10.1051/m2an/2016055

[10] Hao Chen, Xiaoli Wang and Xiaolin Li (2019), doi:10.1016/j.amc.2019.01.041

[11] Md. Masud Rana, Victoria E. Howle, Katharine Long, Ashley Meek and William Milestone (2021), doi:10.1137/20M1349680

[12] Patrick E. Farrell, Robert C. Kirby and Jorge Marchena-Menendez (2021), doi:10.1145/3466168

[13] Xiangmin Jiao, Xuebin Wang and Qiao Chen (2021), doi:10.1137/20M1387985

[14] Yasuhiro Takei and Yoritaka Iwata (2022), doi:10.3390/axioms11010028

[15] Ben S. Southworth, Oliver Krzysik, Will Pazner and Hans De Sterck (2022), doi:10.1137/21M1389742

[16] Ben S. Southworth, Oliver Krzysik and Will Pazner (2022), doi:10.1137/21M1390438

[17] Martin J. Gander and Michal Outrata (2023), doi:10.1016/j.laa.2023.07.008

[18] Shishun Li, Jing-Yuan Wang and Xiao-Chuan Cai (2023), doi:10.1137/22M152880X

[19] Owe Axelsson, Ivo Dravins and Maya Neytcheva (2023), doi:10.1002/nla.2532

[20] Robert C. Kirby (2024), doi:10.1137/23M1569344

[21] Martin J. Gander and Michal Outrata (2024), doi:10.1137/23M1604266

[22] Santolo Leveque, Luca Bergamaschi, Angeles Martinez and John W. Pearson (2024), doi:10.1137/23M1567862

References:
[1] Evans, Lawrence C. (1998). Partial Differential Equations, Number 19. American Mathematical Society.
[2] Hackbusch, Wolfgang. (1994). Iterative Solution of Large Sparse Systems of Equations, Number 95. Springer Verlag.
[3] Hairer, E., S.P. Nørsett, G. Wanner. (1992). Solving Ordinary Differential Equations I - Nonstiff Problems, Springer Verlag, 2nd edition.
[4] Hairer, E., G. Wanner, (1996). Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic Problems, Springer Verlag, 2nd edition.
[5] Hairer, Ernst Gerhard Wanner, (1999). Stiff differential equations solved by Radau methods, Journal of Computational and Applied Mathematics, 111:93-111 doi:10.1016/S0377-0427(99)00134-X
[6] Dennis Jr, J.E., R.B. Schnabel, (1989). A View of Unconstrained Optimization, In G.L. Nemhauser, A.H.G. Rinnooy Kan, and H.J. Todd, editors, Optimization, pages 1-72. Elsevier.
[7] Lagarias, J.C., J.A. Reeds, M.H. Wright, P.E. Wright, (1998). Convergence properties of the Nelder-Meas Simplex method in lower dimensions, SIAM Journal of Optimization, 9:112-147 doi:10.1137/S1052623496303470
[8] Van lent, J. S. Vandewalle, (2004). Multigrid methods for implicit Runge-Kutta and boundary value method discretizations of PDEs, To appear in SIAM J. Sci. Comput.

BibTeX:
@article{MIC-2006-2-3,
  title={{Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs}},
  author={Staff, Gunnar A. and Mardal, Kent-Andre and Nilssen, Trygve K.},
  journal={Modeling, Identification and Control},
  volume={27},
  number={2},
  pages={109--123},
  year={2006},
  doi={10.4173/mic.2006.2.3},
  publisher={Norwegian Society of Automatic Control}
};

Article

“Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs”