Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations

Jackiewicz, Zdzisław; Tracogna, S.

doi:10.1007/bf02142812

Cited by 38 publications

(15 citation statements)

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“…The special case of collocation methods (1.2) provide continuous approximation to the solution y(t) of (1.1) on the whole interval of integration, and not only at the gridpoints {t n } as is the case for the methods defined by (1.3). Different approach to the construction of continuous two-step RungeKutta methods is presented in [4,15] and Bartoszewski et al (unpublished manuscript). Continuous two-step Runge-Kutta methods for delay differential equations are considered in [2,5] and for Volterra integral equations in [10].…”

Section: Introductionmentioning

confidence: 98%

Two-step almost collocation methods for ordinary differential equations

et al. 2009

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Section: Introductionmentioning

confidence: 98%

Two-step almost collocation methods for ordinary differential equations

et al. 2009

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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%

“…A different approach to interpolation which utilizes TSRK methods of order p and stage order q = p has been discussed in [5], [19], [23] and [24].…”

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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“…Jackiewicz and Tracogna [88] developed methods on non-uniform meshes, however this requires the coefficients to be computed every time the stepsize is changed. Tracogna [113] represented the two-step Runge-Kutta methods in Nordsieck form and provided them with local error estimates and interpolants.…”

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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Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations

Cited by 38 publications

References 9 publications

Two-step almost collocation methods for ordinary differential equations

Two-step almost collocation methods for ordinary differential equations

Derivation and implementation of Two-Step Runge-Kutta pairs

General Linear Methods with Inherent Runge‐Kutta Stability

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