Parallel iteration of high-order Runge-Kutta methods with stepsize control

Houwen, P. J. van der; Sommeijer, B. P.

doi:10.1016/0377-0427(90)90200-j

Cited by 130 publications

(72 citation statements)

References 11 publications

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“…Hence, on a parallel computer these RKN methods are expected to be more efficient. In [15], similar type of methods have been constructed for nonstiff first-order ODEs. In that paper, the effective number of stages equals the order.…”

Section: Bp Sommeijer /Explicit Runge-kutta-nystrom Methodsmentioning

confidence: 99%

“…The error control that we have implemented is analogous to the one we have used in our code PIRK for first-order ODEs (this code is given in the appendix to [15]). This error control is essentially based on the observation that the order of the iterates yu> in (2.4a) increases by 2 in each iteration and that substitution of yen into the final line of the RKN methods (i.e., into (2.3a)), yields a reference solution zn+l of order min{p, 2j + q} (cf.…”

Section: Variable-step Implementationmentioning

confidence: 99%

“…The information needed to construct such a prediction could be obtained from approximations calculated in the preceding step. For example, in [17], van der Houwen and Nguyen huu Cong analyse (for first-order ODEs) a block version of the PIRK method [15] with the aim to construct a high-order prediction. The idea is to apply a PIRK method (which is very similar to a PIRKN method) not only with stepsize h, but, in addition (and simultaneously) with stepsizes h; = a;h, i = 1, ... , r -1.…”

Section: The Order Of the Predictormentioning

confidence: 99%

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Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

Sommeijer¹

1993

Applied Numerical Mathematics

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Sommeijer, B.P., Explicit, high-order Runge-Kutta-Nystrom methods for parallel computers, Applied Numerical Mathematics 13 (1993) 221-240.The paper describes the construction of explicit Runge-Kutta-Nystri:im (RKN) methods of arbitrarily high order. The order is borrowed from an underlying implicit RKN method. For the approximate solution of this method, an iteration scheme is defined. Prescribing a fixed number of iterations, the resulting scheme is an explicit RKN method. The iteration scheme is defined in such a way that many of the right-hand side evaluations can be done concurrently. As a result, explicit RKN schemes of order p are obtained which require, on a parallel computer, approximately p /2 right-hand side evaluations per step. Both in fixed-and variable-step mode, the schemes are compared with existing (sequential) high-order RKN methods from the literature and are shown to demonstrate superior behaviour.

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Section: Bp Sommeijer /Explicit Runge-kutta-nystrom Methodsmentioning

confidence: 99%

Section: Variable-step Implementationmentioning

confidence: 99%

Section: The Order Of the Predictormentioning

confidence: 99%

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Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

Sommeijer¹

1993

Applied Numerical Mathematics

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“…This approach was followed in, e.g., Norsett and Simonsen [21], Lie [18], and van der Houwen and Sommeijer [12]. These papers deal with the iteration of implicit methods for solving nonstiff ODE's.…”

Section: Pdirk Methodsmentioning

confidence: 99%

“…First results based on the PC approach were reported by Lie [18], who uses a fourth-order, two-stage Gauss-Legendre corrector and a third-order Hermite extrapolation predictor. In [12], these "parallel, iterated" RK methods (which we shall briefly call PIRK methods) have been investigated for a variety of predictor methods and it was concluded that, from an implementational point of view, one-step predictors are preferable. Related PC methods were studied by Tarn in his thesis [24].…”

Section: Introductionmentioning

confidence: 99%

Embedded diagonally implicit Runge-Kutta algorithms on parallel computers

Houwen¹,

Sommeijer²,

Couzy³

1992

Math. Comp.

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Abstract. This paper investigates diagonally implicit Runge-Kutta methods in which the implicit relations can be solved in parallel and are singly diagonalimplicit on each processor. The algorithms are based on diagonally implicit iteration of fully implicit Runge-Kutta methods of high order. The iteration scheme is chosen in such a way that the resulting algorithm is ^(a)-stable or Z,(a)-stable with a equal or very close to n/2. In this way, highly stable, singly diagonal-implicit Runge-Kutta methods of orders up to 10 can be constructed. Because of the iterative nature of the methods, embedded formulas of lower orders are automatically available, allowing a strategy for step and order variation.

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Load balancing schemes for extrapolation methods

Rauber

Rünger

1997

Concurrency: Pract. Exper.

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Parallel iteration of high-order Runge-Kutta methods with stepsize control

Cited by 130 publications

References 11 publications

Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

Embedded diagonally implicit Runge-Kutta algorithms on parallel computers

Load balancing schemes for extrapolation methods

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