“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.

PDF 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
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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}
};