Parallelism across the steps in iterated Runge-Kutta methods for stiff initial value problems

Houwen, P. J. van der; Sommeijer, B. P.; Veen, W. A. van der

doi:10.1007/bf02142695

Cited by 8 publications

(12 citation statements)

References 11 publications

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“…This recursion is of a similar form as the error recursion of the PDIRKAS GS method analysed in [11] and can be represented as…”

Section: From (24) and (12) It Follows That For Linear Problems Thementioning

confidence: 99%

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Houwen¹,

Veen²

1997

Advances in Computational Mathematics

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Waveform relaxation methods for implicit differential equationsvan der Houwen, P.J.; van der Veen, W.A. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying RungeKutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initialvalue problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.

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“…This recursion is of a similar form as the error recursion of the PDIRKAS GS method analysed in [11] and can be represented as…”

Section: From (24) and (12) It Follows That For Linear Problems Thementioning

confidence: 99%

Untitled

Houwen¹,

Veen²

1997

Advances in Computational Mathematics

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“…For RK-based correction formulas, a convergence analysis of stepparallel methods described above can .be found in [17,18].…”

Section: Jacobi-type Correction Formulamentioning

confidence: 99%

Parallel iteration schemes for implicit ODEIVP methods

Houwen

1993

Seminario Mat. e. Fis. di Milano

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“…Our starting point is the same corrector formula as in [12]. Using the General Linear Method notation of Butcher, the corrector formula reads (cf.…”

Section: The Iteration Schemementioning

confidence: 99%

“…This iteration method was called the PDIRK iteration method (Parallel Diagonalimplicit Iterated RK method). Following the ideas of Bellen and co-workers, a further level of parallelism was introduced in [11][12][13] by applying the PDIRK iteration scheme concurrently at a number of step points on the t-axis. In this paper, we shall analyse the convergence of these step-parallel PDIRK methods.…”

Section: Y'(t)=f(y(t)) Y(to)=yo Yf~dmentioning

confidence: 99%

“…In addition to parallelism across the steps, PIRKAS methods also have parallelism across the components of the iterates, because all components of F(¥~ j-~)) can also be evaluated in parallel. Methods where B is a diagonal matrix D with positive diagonal entries minimizing the spectral radius of the matrix I -D-IA (such matrices can be found in [10]) were applied in [12] and were called PDIRKAS methods (Parallel Diagonal-implicitly Iterated RK Across the Steps). These methods are implicit because we have to solve nonlinear relations in each iteration.…”

Section: )mentioning

confidence: 99%

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