Order conditions for RKN methods solving general second-order oscillatory systems

“…The Runge-Kutta type methods [11][12][13][14][15][16][17][18] that usually solve these second-order systems require a large number of evaluations of the perturbation function. This is an inconvenience for its implementation on a computer.…”

Section: Introductionmentioning

confidence: 99%

An Algorithm for the Numerical Integration of Perturbed and Damped Second-Order ODE Systems

2020

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A new method of numerical integration for a perturbed and damped systems of linear second-order differential equations is presented. This new method, under certain conditions, integrates, without truncation error, the IVPs (initial value problems) of the type: x″(t)+Ax′(t)+Cx(t)=εF(x(t),t), x(0)=x0, x′(0)=x0′, t∈[a,b]=I, which appear in structural dynamics, astrodynamics, and other fields of physics and engineering. In this article, a succession of real functions is constructed with values in the algebra of m×m matrices. Their properties are studied and we express the solution of the proposed IVP through a serial expansion of the same, whose coefficients are calculated by means of recurrences involving the perturbation function. This expression of the solution is used for the construction of the new numerical method. Three problems are solved by means of the new series method; we contrast the results obtained with the exact solution of the problem and with its first integral. In the first problem, a quasi-periodic orbit is integrated; in the second, a problem of structural dynamics associated with an earthquake is studied; in the third, an equatorial satellite problem when the perturbation comes from zonal harmonics J2 is solved. The good behavior of the series method is shown by comparing the results obtained against other integrators.

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“…In the last decades, several authors have given a great attention on developing RKN methods (see [5,6,7,8,10,11]).…”

Section: Introductionmentioning

confidence: 99%

A NEW IMPROVED RUNGE-KUTTA FORMULA FOR DIRECTLY SOLVING $z''(t)=g(t,z,z')$

Hussain¹,

Ismail²

2020

Int. J. of Appl. Math.

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This paper deals with the derivation of an explicit two-stage thirdorder Improved Runge-Kutta Nyström (IRKNG) method for directly solving general second order ordinary differential equations (ODE). This method is twostep and the number of functions to be evaluated per step is less via comparsion to the existing RK methods. Numerical outcomes are offered to show the validity and competency of the newly IRKNG method as comparison with the general RKN and RK techniques.

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Order conditions for RKN methods solving general second-order oscillatory systems

Cited by 33 publications

References 17 publications

A Class of Explicit Divergence-Free Methods for Maxwell’s Equations With Dirichlet Boundary Conditions

A Class of Explicit Divergence-Free Methods for Maxwell’s Equations With Dirichlet Boundary Conditions

An Algorithm for the Numerical Integration of Perturbed and Damped Second-Order ODE Systems

A NEW IMPROVED RUNGE-KUTTA FORMULA FOR DIRECTLY SOLVING $z''(t)=g(t,z,z')$

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