An implicit one-step method of high-order accuracy for the numerical integration of ordinary differential equations

Urabe, Minoru

doi:10.1007/bf02165379

Cited by 27 publications

(15 citation statements)

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“…Recently Urabe [8] has developed a high order, one step algorithm for the solution of systems of the form (1.1), which is efficient in the case where the time constants associated with (1.1) are all large. In this paper we extend Urabe's method to deal with systems of the form (1.1) which have widely separated time constants all of which are positive i.e.…”

Section: J(t)=f~(t P(t))mentioning

confidence: 99%

High order methods for the numerical integration of ordinary differential equations

Cash

1978

Numer. Math.

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Summary. High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.Subject Classifications. AMS(MOS): 65 L05; CR: 5.17.

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Section: J(t)=f~(t P(t))mentioning

confidence: 99%

High order methods for the numerical integration of ordinary differential equations

Cash

1978

Numer. Math.

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“…This means that the last row of the coefficient matrix is identical with the vector of output value coefficients (see, [2]). Many researchers have worked on the second derivative type of numerical integration methods ( see, for example [5], [13,12], [15,16] and [17]) for approximating the solution of initial value problem in ordinary differential equations (ODEs) of the form y = f (x, y(x)), a ≤ x ≤ b, y(x 0 ) = y 0 , y :…”

Section: Introductionmentioning

confidence: 99%

Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems

Yakubu

Kumleng

Markuš

2017

J. of Mod. Meth. in Numer. Math.

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Second derivative Runge-Kutta collocation methods for the numerical solution of stiff system of first order initial value problems in ordinary differential equations are derived and studied. The inclusion of the second derivative terms enabled us to derive a set of methods which are convergent with large regions of absolute stability. Although the implementation of the methods remains iterative in a precisely defined way, the advantage gained makes them suitable for solving stiff system of equations with large Lipschitz constants. The derived methods are illustrated by the applications to some test problems of stiff system found in the literature and the numerical results obtained confirm the potential of the second derivative methods.

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“…In this paper we introduce second-derivative of high-order accuracy methods with off-step points designed for the numerical integration of such systems of initial value problems. Although several authors have studied methods with second derivative terms, for example, see [7,16,21,22,[26][27][28]. Further, some of the methods considered for the solution of (1.1) were derived on the basis that the required function evaluations are to be done only at the grid points (discrete points) which is typically of discrete variable methods [18] (Euler method, Runge-Kutta methods, Picard method, etc.).…”

Section: Introductionmentioning

confidence: 99%

Second derivative of high-order accuracy methods for the numerical integration of stiff initial value problems

Yakubu

Markuš

2016

Afr. Mat.

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In Mitsui (Sci Eng Rev Doshisha Univ Jpn 51(3): [181][182][183][184][185][186][187][188][189][190] 2010) the author introduced a new class of discrete variable methods known as look-ahead linear multistep methods (LALMMs) which consist of a pair of predictor-corrector (PC), including a function value at one more step beyond the present step for the numerical solution of ordinary differential equations. Two-step family of Look-Ahead linear multistep methods of fourth-order pair were derived and shown to be A(θ )-stable in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181-188, 2011). The derived integration methods are of low orders and unfortunately cannot cope with stiff systems of ordinary differential equations. In this paper we extend the concept adopted in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181-188, 2011) to construct second-derivative of high-order accuracy methods with off-step points which behave essentially like one-step methods. The resulting integration methods are A-stable, convergent, with large regions of absolute stability, suitable for stiff systems of ordinary differential equations. Numerical comparisons of the new methods have been made and enormous gains in efficiency are achieved.

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An implicit one-step method of high-order accuracy for the numerical integration of ordinary differential equations

Cited by 27 publications

References 1 publication

High order methods for the numerical integration of ordinary differential equations

High order methods for the numerical integration of ordinary differential equations

Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems

Second derivative of high-order accuracy methods for the numerical integration of stiff initial value problems

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