High order methods for the numerical integration of ordinary differential equations

Cash, J. R.

doi:10.1007/bf01398507

Cited by 13 publications

(8 citation statements)

References 4 publications

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“…i. R. CASH Scheme (4.1), which is of the type considered for nonstiff equations in [1], is not even zero-stable, owing to the fact that there are two roots of the polynomial equation…”

Section: Krmentioning

confidence: 99%

An extension of Olver’s method for the numerical solution of linear recurrence relations

Cash¹

1978

Math. Comp.

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An algorithm is developed for computing the solution of a class of linear recurrence relations of order greater than two when unstable error propagation prevents the required solution being found by direct forward recurrence. By abandoning an appropriate number of initial conditions the original problem may be replaced by an inexact but well-conditioned boundary value problem, and in certain circumstances the solution of this new problem is a good approximation to the required solution of the original problem. The required solution of this reposed problem is generated using an algorithm based on Gaussian elimination, and a technique developed by Olver is extended to estimate automatically the truncation error of the proposed algorithm.

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“…i. R. CASH Scheme (4.1), which is of the type considered for nonstiff equations in [1], is not even zero-stable, owing to the fact that there are two roots of the polynomial equation…”

Section: Krmentioning

confidence: 99%

An extension of Olver’s method for the numerical solution of linear recurrence relations

Cash¹

1978

Math. Comp.

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“…where /, =f(x 0 + ih, yj) 9 g t =f(x 0 + ih, yj) 9 f t =f(x Q + h, j> 0 and g 1 = g(x 0 + h, j>0-Succeeding his result, J. R. CASH [5] has considered this type of formula more generally and made some stability analysis. On the other hand H. SHIN-TANI [12], [13] has proposed some formulas analogous to RK formula employing one evaluation for /(x, j;) and some for g (x, y).…”

Section: Introductionmentioning

confidence: 96%

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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“…Second, a substantial increase in efficiency may be achieved by the numerical integration methods which utilize the second-derivative terms. Third, the relatively good stability properties enjoyed by these methods make them more efficient for the numerical integration of systems having Jacobians with eigenvalues lying close to the imaginary axis (see Cash [13]). …”

Section: Introductionmentioning

confidence: 99%

Implicit two-derivative Runge–Kutta collocation methods for systems of initial value problems

Yakubu

Kwami

2015

Journal of the Nigerian Mathematical Society

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We introduce a new class of implicit two-derivative Runge-Kutta collocation methods designed for the numerical solution of systems of equations and show how they have been implemented in an efficient parallel computing environment. We also discuss the difficulty associated with large systems and how, in this case, one must take advantage of the second derivative terms in the methods. We consider two modified versions of the methods which are suitable for solving stable systems. The first modification involves the introduction of collocation at the two end points of the integration interval in addition to the Gaussian interior collocation points and the second involves the introduction of a different class of basic second derivative methods. With these modifications, fewer function evaluations per step are achieved, resulting into methods that are cheap and easy to implement. The stability properties of these methods are investigated and numerical results are given for each of the modified version to illustrate the computational efficiency of the modified methods.

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High order methods for the numerical integration of ordinary differential equations

Cited by 13 publications

References 4 publications

An extension of Olver’s method for the numerical solution of linear recurrence relations

An extension of Olver’s method for the numerical solution of linear recurrence relations

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

Implicit two-derivative Runge–Kutta collocation methods for systems of initial value problems

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