Coefficients for the study of Runge-Kutta integration processes

Butcher, John

doi:10.1017/s1446788700027932

Cited by 363 publications

(219 citation statements)

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“…Thus, The remaining coefficients may be obtained by considering conditions (7.2) and (7.3). These give it has been verified that the coefficients satisfy all the order conditions given by Butcher [1]. A similar method may be obtained by interchanging c3 and c4 and this gives c2 E (0, 1), but some coefficients are large.…”

Section: Pij) Dir)mentioning

confidence: 60%

Semiexplicit 𝐴-stable Runge-Kutta methods

Cooper¹,

Sayfy²

1979

Math. Comp.

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Abstract.An s -1 stage semiexplicit Runge-Kutta method is represented by an î X i real lower triangular matrix where the number of implicit stages is given by the number of nonzero diagonal elements. It is shown that the maximum order attainable is i when s < 5. Necessary and sufficient conditions for ^-stability are derived and it is shown that there must be î -1 implicit stages if the order is î and s < 5. Examples are given for s < 4 where all the nonzero diagonal elements are equal. Additional problems arise when i > 4; but when î = 6, an .4-stable method of order 5 is obtained. This method has five nonzero diagonal elements, and these elements are equal. Finally, a six stage bistable method of order six is given. Again, this method has five nonzero (and equal) diagonal elements.

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Section: Pij) Dir)mentioning

confidence: 60%

Semiexplicit 𝐴-stable Runge-Kutta methods

Cooper¹,

Sayfy²

1979

Math. Comp.

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“…. , c 6 The exact solution of the resulted nonlinear system is out of question. Only after considering simplifying assumptions, we can use nonlinear optimization techniques to get accurate enough solutions.…”

Section: 1mentioning

confidence: 99%

On the modification of Differential Evolution strategy for the construction of Runge Kutta pairs

Famelis

Tsitouras²

2016

MATEC Web of Conferences

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Abstract. Among the most popular methods for the solution of the Initial Value Problem are the Runge-Kutta (RK) pairs. These methods can be derived by solving a system of nonlinear equations after admitting various simplifying assumptions. The more simplifying assumptions we consider the more we narrow the solution space. In [1] Tsitouras presented an algorithm for the construction of Runge-Kutta pairs of orders 5 and 4 satisfying only the so called "first column simplifying assumption". In [2] Famelis and Tsitouras have studied the ability of Differential Evolution techniques to find solutions satisfying all the order conditions needed for the derivation of orders 5 and 4 pairs. In this work we propose an modification on the Differential Evolution strategy for the same problem. The current study will be a guide to the construction of other classes of RK that have not been presented in the literature.

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“…fj -, \i { we have many variations, among which the classical Runge-Kutta method or the Runge-Kutta-Fehlberg method is famous. A distinguished contribution for the study of the RungeKutta methods has been made by J. C. BUTCHER ( [1] ~ [4]). He determined the attainable order of the RK methods up to 10-stage formula.…”

Section: Introductionmentioning

confidence: 99%

Runge-Kutta Type Integration Formulas Including the Evaluation of the Second Derivative, Part I

Mitsui

1982

Publ. Res. Inst. Math. Sci.

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Coefficients for the study of Runge-Kutta integration processes

Cited by 363 publications

References 3 publications

Semiexplicit 𝐴-stable Runge-Kutta methods

Semiexplicit 𝐴-stable Runge-Kutta methods

On the modification of Differential Evolution strategy for the construction of Runge Kutta pairs

Runge-Kutta Type Integration Formulas Including the Evaluation of the Second Derivative, Part I

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