Abstract
Peer two-step W-methods are designed for integration of stiff initial value problems with parallelism across the method. The essential feature is that in each time step s ‘peer’ approximations are employed having similar properties. In fact, no primary solution variable is distinguished. Parallel implementation of these stages is easy since information from one previous time step is used only and the different linear systems may be solved simultaneously. This paper introduces a subclass having order s−1 where optimal damping for stiff problems is obtained by using different system parameters in different stages. Favourable properties of this subclass are uniform stability for realistic stepsize sequences and a superconvergence property which is proved using a polynomial collocation formulation. Numerical tests on a shared memory computer of a matrix-free implementation with Krylov methods are included.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Brugnano and C. Magherini, Blended implementation of block implicit methods for ODEs, Appl. Numer. Math., 42 (2002), pp. 29–45.
K. Burrage and T. Tian, Parallel half-block methods for initial value problems, Appl. Numer. Math., 32 (2000), pp. 255–271.
J. C. Butcher, General linear methods for stiff differential equations, BIT, 41 (2001), pp. 240–264.
G. D. Byrne, Pragmatic experiments with Krylov methods in the stiff ODE setting, in Computational Ordinary Differential Equations, Clarendon Press, Oxford, 1992, pp. 323–356.
G. Dahlquist and A. Björck, Numerical Methods, Dover Publications, 1974.
J. L. Frank and P. J. van der Houwen, Parallel iteration of the extended backward differentiation formulas, IMA J. Numer. Anal., 21 (2001), pp. 367–385.
W. Gautschi and G. Inglese, Lower bounds for the condition number of Vandermonde matrices, Numer. Math., 52 (1988), pp. 241–250.
W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations, Springer Series in Computational Mathematics 33, Springer, Berlin, 2003.
Y. Saad, Iterative Methods for Sparse Linear systems, PWS Publishing Company, Boston, 1996.
B. A. Schmitt and R. Weiner, Parallel two-step W-methods with Peer variables, SIAM J. Numer. Anal., 42 (2004), pp. 265–282.
B. A. Schmitt, R. Weiner, and K. Erdmann, Implicit parallel peer methods for stiff initial value problems, to appear in Appl. Numer. Math.
B. P. Sommeijer, L. F. Shampine, and J. G. Verwer, RKC: An explicit solver for parabolic PDEs, J. Comp. Appl. Math., 88 (1997), pp. 315–326.
R. Weiner, B. A. Schmitt, and H. Podhaisky, Parallel two-step W-methods on singular perturbation problems, in SLNCS 2328, 2002, pp. 778–785.
R. Weiner, B. A. Schmitt, and H. Podhaisky, Parallel ‘Peer’ two-step W-methods and their application to MOL systems, Appl. Numer. Math., 48 (2004), pp. 425–439.
R. Weiner, B. A. Schmitt, and H. Podhaisky, ROWMAP – a ROW-code with Krylov techniques for large stiff ODEs, Appl. Numer. Math., 25 (1997), pp. 303–319.
J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS subject classification (2000)
65L06, 65Y05.
Received June 2004. Revised January 2005. Communicated by Timo Eirola.
Helmut Podhaisky: The work of this author was supported by the German Academic Exchange Service, DAAD.
Rights and permissions
About this article
Cite this article
Schmitt, B.A., Weiner, R. & Podhaisky, H. Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration. Bit Numer Math 45, 197–217 (2005). https://doi.org/10.1007/s10543-005-2635-y
Issue Date:
DOI: https://doi.org/10.1007/s10543-005-2635-y