Parallel Two-Step W-Methods with Peer Variables

Schmitt, Bernhard A.; Weiner, Rüdiger

doi:10.1137/s0036142902411057

Cited by 66 publications

(51 citation statements)

References 14 publications

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“…As it has been pointed out in [11], [18] the method (5) is zero-stable if and only if the matrix A has an eigenvalue λ 1 (A) = 1 and the remaining eigenvalues λ j (A), j = 2, . .…”

Section: Introductionmentioning

confidence: 93%

On the derivation of explicit two-step peer methods

Calvo

Montijano

Rández

et al. 2011

Applied Numerical Mathematics

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The so called peer methods for the numerical solution of Initial Value Problems (IVP) in ordinary differential systems were introduced and later applied to different types of problems by R. The aim of this paper is to propose an alternative procedure to construct families of explicit two step peer methods in which the available parameters appear in a transparent way. This allows us to obtain families of fixed stepsize s stage methods with stage order 2s−1, which provide dense output without extra cost, depending on some free parameters that can be selected taking into account the stability properties and leading error terms. A study of the extension of these methods to variable stepsize is also carried out. Optimal s stage methods with s = 2, 3 are derived.

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“…As it has been pointed out in [11], [18] the method (5) is zero-stable if and only if the matrix A has an eigenvalue λ 1 (A) = 1 and the remaining eigenvalues λ j (A), j = 2, . .…”

Section: Introductionmentioning

confidence: 93%

On the derivation of explicit two-step peer methods

Calvo

Montijano

Rández

et al. 2011

Applied Numerical Mathematics

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“…Here, we consider explicit two-step peer methods introduced by R. Weiner, et al [6,7] as an alternative to classical Runge-Kutta (RK) and multistep methods attempting to combine the advantages of these two classes of methods.…”

Section: Introductionmentioning

confidence: 99%

“…

Abstract.The so called peer methods for the numerical solution of Initial Value Problems (IVP) in ordinary differential systems were introduced by R. Weiner et al [6,7,11,12,13] for solving different types of problems either in sequential or parallel computers. In this work, we study exponentially fitted three-stage peer schemes that are able to fit functional spaces with dimension six.

…”

mentioning

confidence: 99%

Exponentially fitted fifth-order two-step peer explicit methods

Calvo¹,

Montijano²,

Rández³

et al. 2015

AIP Conference Proceedings

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“…y(t 0 ) = y 0 ∈ R n , (1.1) [10]. These methods have an inherent parallelism across the method since they employ s stages which are totally independent within the actual time step.…”

Section: Introductionmentioning

confidence: 99%

“…As an example, singly-implicit methods being almost L-stable exist up to order 7 [10]. Also, the methods do not suffer from order reduction in stiff problems [11].…”

Section: Introductionmentioning

confidence: 99%

Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration

Schmitt

Weiner²,

Podhaisky

2005

Bit Numer Math

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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. (2000): 65L06, 65Y05. AMS subject classification

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Parallel Two-Step W-Methods with Peer Variables

Cited by 66 publications

References 14 publications

On the derivation of explicit two-step peer methods

On the derivation of explicit two-step peer methods

Exponentially fitted fifth-order two-step peer explicit methods

Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration

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