Variable-Stepsize Explicit Two-Step Runge-Kutta Methods

Jackiewicz, Z.; Zennaro, M.

doi:10.2307/2153065

Cited by 6 publications

(6 citation statements)

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“…the function f per step and hence are not as efficient as, for example, linear multistep methods, when the derivative evaluations are relatively expensive. To seek compromises between the strengths and weaknesses of the standard methods, a number of authors [4], [5], [7], [8], [9], [10], [12], [14], [15], [16], [17], [18], [19], [20], [23], [24] have studied the possibility of using approximations to the solution and its derivatives at two consecutive steps. This approach leads to the general class of two-step Runge-Kutta (TSRK) methods of the form Y' = ujyi_l + (1 -uj)yi f h^(ajkf(Yk 1) + bjkf(Yik )) , implementations exploit explicit RK pairs for efficiency, and so the focus here is on the derivation of TSRK pairs suitable for implementation with variable stepsizes.…”

Section: Introductionmentioning

confidence: 99%

Derivation and implementation of Two-Step Runge-Kutta pairs

Jackiewicz

Verner

2002

Japan J. Indust. Appl. Math.

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Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two-step Runge-Kutta methods strive to improve the efficiency by utilizing approximations to the solution and its derivatives from the previous step. This article suggests a strategy for computing embedded pairs of such two-step methods using a smaller number of function evaluations than that required for traditional Runge-Kutta methods of the same order. This leads to the efficient and reliable estimation of local discretization error and a robust step control strategy. The change of stepsize is achieved by a suitable interpolation of stage values from the previous step and does not require any additional function evaluations. Two examples illustrate the features of these pairs.

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Section: Introductionmentioning

confidence: 99%

Derivation and implementation of Two-Step Runge-Kutta pairs

Jackiewicz

Verner

2002

Japan J. Indust. Appl. Math.

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“…These methods were introduced by Renaut [106], Jackiewicz, Renaut and Feldstein [85] and Jackiewicz, Renaut and Zennaro [86]. These methods were generalised by Jackiewicz and Tracogna [87] and are given by…”

Section: Two Step Runge-kutta Methodsmentioning

confidence: 99%

General Linear Methods with Inherent Runge‐Kutta Stability

Jackiewicz¹

2009

General Linear Methods for Ordinary Differential Equations

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General linear methods were derived approximately thirty years ago as a unifying approach for the study of consistency, stability and convergence of the Runge-Kutta and the linear multistep methods. Their discovery opened the possibility of obtaining essentially new methods which were neither Runge-Kutta nor linear multistep methods nor slight variations of these methods. It was hoped that general linear methods would exist which are practical and have advantages over the traditional methods. Locating such practical methods has proved difficult though, for several reasons. For example, the complexity of the order conditions becomes very high, making it difficult to even find the required conditions in many cases let alone solve them.Several simplifying assumptions must be made, which limit this large class to situations where practical methods are likely to exist. The first assumption is that the stage order is equal to the overall order of the general linear method. This results in methods which, among other things, are not affected by the order reduction phenomenon. The second assumption is that a Nordsieck vector is passed from step to step. This enables such methods to vary the stepsize in a convenient and practical way. The final assumption is that the stability regions of the general linear methods should be identical to those of corresponding Runge-Kutta methods.The first two assumptions are satisfied by the structure of the U and V matrices of the general linear methods. In order to satisfy the last assumption sufficient conditions are developed. These conditions result in a class of general linear methods with a property known as inherent RungeKutta stability (IRKS). The IRKS conditions relate the coefficient matrices of the general linear method with a doubly companion matrix X to satisfyConstructing general linear methods with the IRKS property in the most general way possible is the main aim of this thesis. To derive these methods a transformation is used; this transformation brings all methods of this class into a particular form, which allows the construction using only linear operations.Several special properties of methods with the IRKS property are introduced. For example, conditions which show that the ESIRK methods are a special case of the IRKS methods are introduced. This then allows the introduction of a new class of ESIRK methods which may have advantages over those already known. Also, methods which have a property known as strong stiff accuracy are developed which make them similar to strictly stable Runge-Kutta methods. Methods with strong stiff accuracy are likely to be considered good, particularly because they are the most suitable amongst the IRKS methods for the solution of differential algebraic equations.The theoretical properties of the general linear methods with IRKS are investigated using various implementations from fixed stepsize and fixed order to variable stepsize and variable order codes. The IRKS methods are experimentally compared with several traditional metho...

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“…It must be noticed that integration methods that combine information about the solution in two consecutive steps have been considered for a long time as can be seen in the earlier publications [2], [7], [8] and [16]. In particular, the General Linear Methods introduced by J.C. Butcher in [2] as a generalization of linear multistep (multivalue) methods and Runge-Kutta (multistage) methods also contain the explicit peer two-step methods.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, even though not all matrices A satisfying the spectral assumption (7) can be written in the form (8), (9), those for which the corresponding S admits a LU decomposition without pivoting can.…”

Section: Introductionmentioning

confidence: 99%

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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Variable-Stepsize Explicit Two-Step Runge-Kutta Methods

Cited by 6 publications

References 7 publications

Derivation and implementation of Two-Step Runge-Kutta pairs

Derivation and implementation of Two-Step Runge-Kutta pairs

General Linear Methods with Inherent Runge‐Kutta Stability

On the derivation of explicit two-step peer methods

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