Explicit, two‐stage, sixth‐order, hybrid four‐step methods for solving

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Section: Numerical Resultsmentioning

confidence: 99%

“…Recently, a similar approach was given in as a variant of Runge–Kutta–Nyström type methods with fixed b 1 = −2, b 2 = 2, b 3 = −2, and b 4 = 1. A little later a general constant coefficients family of this type of methods was studied and b 's are freely chosen within the stability limitations …”

Section: Introductionmentioning

confidence: 99%

Trigonometric–fitted hybrid four–step methods of sixth order for solving

Медведев

Simos

Tsitouras³

2018

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“…Li and Wang presented an extension of Runge‐Kutta‐Nyström type methods that was actually a special case of . The present authors, introduced a four‐step family of methods attaining sixth algebraic order . Subsequently, a variable coefficient sixth order, hybrid four‐step method was appeared in the study of Medvedev, Simos, and Tsitouras …”

Section: Preliminariesmentioning

confidence: 99%

Hybrid, phase–fitted, four–step methods of seventh order for solving x″(t) = f(t,x)

Медведев

Simos

Tsitouras³

2019

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A four‐step method of seventh algebraic order is presented. It is tuned for addressing the special second order initial value problem. The new method is hybrid, explicit, and uses three stages per step. In addition is phase fitted. In consequence it uses variable coefficients that depend on the magnitude of the step‐size. We also present numerical tests on a set of standard problems that illustrate the efficiency of the derived method over older ones given in the relevant literature.

“…The sixth order explicit methods chosen to be tested are: The six‐step method presented here named for simplicity SS6p. The six‐step method of Lambert & Watson named LW6. The four‐step method with phase lag O ( v 12 ) given in Medvedev et al named FS6p. The two‐step method presented in Tsitouras named TS6. The RKN pair with phase lag O ( v 10 ) given in Papakostas named RKN6(4)p. …”

Section: Numerical Performancesmentioning

confidence: 99%

“…A first choice is the test equation problem

y^{″} (t) = - 25 y (t), y (0) = 0, y^{″} (0) = 5,

with theoretical solution

y (t) = \sin (5 t) .

We integrated this problem for

t \in ([], 0, 20 π) .

The results are presented in Table .…”

Section: Numerical Performancesmentioning

confidence: 99%

Explicit hybrid six–step, sixth order, fully symmetric methods for solving y ″ = f (x,y)

Fang

Liu

Hsu

et al. 2019

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A family of explicit, fully symmetric, sixth order, six‐step methods for the numerical solution of y′′ = f(x,y) is studied. This family wastes two function evaluations per step and can be derived through interpolation techniques. An interval of periodicity is possessed and the phase lag is of high order. Numerical instabilities usually present in such type of multistep methods were circumvented. We conclude with extended numerical tests over a set of problems justifying our effort of dealing with the new methods.