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

Self Cite

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.

Section: Preliminariesmentioning

confidence: 99%

“…In the study of Lambert and Watson, it was proposed the scalar model equation

x^{″} = - ω^{2} x, ω \in double-struckR,

for studying the solution of when periodic solutions are present. () As in the studies of Medvedev, Simos, and Tsitouras and Simos and Tsitouras,() we may consider for simplicity,

x false(0 false) = 1, x^{'} false(0 false) = 0 .

…”

Section: Problems With Periodic Solutionsmentioning

confidence: 99%

Section: Numerical Performancementioning

confidence: 99%

See 1 more Smart Citation

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

Медведев

Simos

Tsitouras³

2019

Self Cite

“…We performed our tests with the following sixth‐order methods: The six‐step, phase‐fitted method presented here, named for simplicity SS6ph. The six‐step method SS6p appeared in Fang et al The four‐step, trigonometric‐fitted method FS6t appeared in Medvedev et al …”

Section: Numerical Performancesmentioning

confidence: 99%

“…In the previous studies, various four‐step methods were presented sharing variable coefficients while in other studies, we proposed two‐stage, four‐step methods of order six and three‐stage methods of order seven, respectively. As companion of these four‐step methods, we presented their trigonometric fitted and phase‐fitted counterparts in the previous studies …”

Section: Introductionmentioning

confidence: 99%

Phase‐fitted, six‐step methods for solving x′′ = f(t,x)

Liu

Hsu

Simos

et al. 2019

Self Cite

An explicit, six‐step method of sixth order is presented and tuned for the numerical solution of x′′ = f(t,x). This method is explicit, hybrid, and uses two function evaluations (stages) per step. Its coefficients are varied and depend on the step size. This variance comes from the demand of the method to nullify the phase errors produced when solving the standard simple oscillator. The first and second derivative of this error vanish also. Numerical tests in a set of relevant problems illustrate the efficiency of the newly derived method.

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

Fang

Liu

Hsu

et al. 2019

Self Cite

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.