Explicit two-step methods for second-order linear IVPs

Tsitouras, Ch.

doi:10.1016/s0898-1221(02)80004-9

Cited by 35 publications

(18 citation statements)

References 16 publications

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“…For the problem , it was natural to consider RKN methods. According to discussion in Tsitouras, this is not as effective as it might look at first. Thus, in a previous study, we presented a modified Numerov method of order 7 using 4 stages per step.…”

Section: Introductionmentioning

confidence: 97%

Evolutionary generation of high‐order, explicit, two‐step methods for second‐order linear IVPs

Simos

Tsitouras²

2017

Math Methods in App Sciences

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In this paper, we consider the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients. Hybrid Numerov methods are used that are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. We present the order conditions taking advantage of the special structure of the problem at hand. These equations are solved using differential evolution technique, and we present a method with algebraic order eighth at a cost of only 5 function evaluations per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation indicate the superiority of the new method.

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Section: Introductionmentioning

confidence: 97%

Evolutionary generation of high‐order, explicit, two‐step methods for second‐order linear IVPs

Simos

Tsitouras²

2017

Math Methods in App Sciences

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“…Zingg and Chisholm presented Runge‐Kutta (RK) methods of orders four, five, and six for first‐order linear initial value problems. Tsitouras presented a hybrid Numerov‐type method of order seven needed four stages per step. This was an improvement since only sixth‐order method can be derived with four stages for Equation , .…”

Section: Introductionmentioning

confidence: 99%

Ninth‐order, explicit, two‐step methods for second‐order inhomogeneous linear IVPs

2020

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In the present paper, the second‐order linear inhomogeneous initial value problems (LI‐IVP) with constant coefficients is of our interest. Two‐step hybrid methods (ie, of Numerov‐type) are considered for addressing this problem. Thus, a special set of order conditions is given and solved for derivation a low cost new method. Actually, we manage to save two stages per step. This is crucial as shown in various numerical tests where our new proposal outperforms standard methods in the relevant literature.

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“…Firstly, Chawla , and Chawla and Rao presented an explicit modification of the classical Numerov method by inserting an extra stage. Since then, several authors have used extra stages to increase the order of the methods . Raptis and Allison considered an exponentially fitted modification of the classical Numerov method.…”

Section: Introductionmentioning

confidence: 99%

Modified two‐step hybrid methods for the numerical integration of oscillatory problems

Monovasilis¹,

Kalogiratou²,

Ramos

et al. 2017

Math Methods in App Sciences

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The construction of modified two‐step hybrid methods for the numerical solution of second‐order initial value problems with periodic or oscillatory behavior is considered. The coefficients of the new methods depend on the frequency of each problem so that the harmonic oscillator y′′=−w2y is integrated exactly. Numerical experiments indicate that the new methods are more efficient than existing methods with constant or variable coefficients. Copyright © 2017 John Wiley & Sons, Ltd.

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Explicit two-step methods for second-order linear IVPs

Cited by 35 publications

References 16 publications

Evolutionary generation of high‐order, explicit, two‐step methods for second‐order linear IVPs

Evolutionary generation of high‐order, explicit, two‐step methods for second‐order linear IVPs

Ninth‐order, explicit, two‐step methods for second‐order inhomogeneous linear IVPs

Modified two‐step hybrid methods for the numerical integration of oscillatory problems

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