Local accuracy and error bounds of the improved Runge-Kutta numerical methods

Qureshi, Sania; Memon, Zaib-un-Nisa; Shaikh, Asif Ali

doi:10.17512/jamcm.2018.4.08

Cited by 3 publications

(3 citation statements)

References 12 publications

(12 reference statements)

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“…Various numerical experiments have been carried out to test the performance of the proposed scheme given in (9). The comparison is made with the fourth-order Taylor series and a nonlinear RK type scheme based upon harmonic mean (RK-HM) [21].…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…It is described in [9][10][11][12][13][14][15] that if initial value problems have singularities and oscillatory solutions, then conventional approaches including single-step Runge-Kutta and linear multistep methods cannot perform well, whereas unconventional approaches can be used to achieve the solution with higher-order numerical accuracy and low computational cost. Such approaches have stronger stability properties as well.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new nonlinear L-stable scheme with constant and adaptive step-size strategy

Arain¹,

Qureshi²,

Shaikh³

2021

J. Appl. Math. Comput. Mech.

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The present study proposes a new explicit nonlinear scheme that solves stiff and nonlinear initial value problems in ordinary differential equations. One of the promising features of this scheme is its fourth-order convergence with strong stability having an unbounded region. A modern approach for relative stability growth analysis is also presented under order stars conditions. The scheme is also good in dealing with singular and stiff type of models. Comparing numerical experiments using various errors, including maximum absolute global error over the integration interval, absolute error at the endpoint, average error, norm of errors, and the CPU times (seconds), shows better performance. An adaptive step-size approach seems to increase the performance of the proposed scheme. The numerical simulations assure us of L -stability, consistency, order, and rapid convergence.

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Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A new nonlinear L-stable scheme with constant and adaptive step-size strategy

Arain¹,

Qureshi²,

Shaikh³

2021

J. Appl. Math. Comput. Mech.

Self Cite

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“…Standard numerical techniques to solve initial value problems in ordinary differential equations include linear explicit and implicit Runge-Kutta techniques, linear explicit and implicit Adams-Bashforth-Moulton schemes, exponential schemes, multiderivative schemes, backward differentiation formulae, and a few others [12]. Among the nonstandard; improved linear explicit Runge-Kutta schemes with reduced slope evaluations, accelerated Runge-Kutta schemes, singly-implicit Runge-Kutta schemes, A-stable Runge-Kutta collocation schemes, two-derivative Runge-Kutta schemes, semi-implicit hybrid schemes, explicit and implicit block schemes, and the list goes on as can be found in [13][14][15][16][17]. Apart from these, nonlinear/rational numerical techniques to solve mathematical models having characteristics of stiffness and singularity have been developed [18].…”

Section: Introductionmentioning

confidence: 99%

Use of partial derivatives to derive a convergent numerical scheme with its error estimates

Qureshi¹,

Adeyeye²,

Shaikh³

2019

J. Appl. Math. Comput. Mech.

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Using the idea of the partial derivative with respect to the ordinate of a given mathematical function, a new numerical scheme having third order convergence has been devised for solving initial value problems in ordinary differential equations. Such problems are deemed to be indispensable in diverse fields of science, medical and engineering and are most often required to be solved by the numerical schemes. In view of this, the proposed numerical scheme is found to be efficient in solving both autonomous and non-autonomous type of problems as supported by some numerical experiments in the present study. Using the Taylor expansion for the slopes involved in the scheme, the leading term of the local truncation error is shown to have contained O(h 4) which proves third order accuracy of the scheme. In addition to this, consistency and linear stability analysis of the proposed scheme has extensively been discussed. Numerical experiments show better performance of the proposed numerical scheme when compared with existing numerical schemes of the same order as that of the scheme proposed. CPU time (seconds), maximum absolute relative error and the absolute relative error, computed at the last grid point of the integration interval for the associated initial value problem, are the parameters to test the performance of the proposed numerical scheme. MATLAB Version: 9.4.0.813654 (R2018a) in double-precision on a personal computer equipped with a Processor Intel (R) Core(TM) i3-4500U CPU@ 1.70 GHz running under the Windows 10 operating system has been employed in order to carry out all the required numerical computations.

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A new third order convergent numerical solver for continuous dynamical systems

Qureshi

Yusuf²

2020

Journal of King Saud University - Science

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Local accuracy and error bounds of the improved Runge-Kutta numerical methods

Cited by 3 publications

References 12 publications

A new nonlinear L-stable scheme with constant and adaptive step-size strategy

A new nonlinear L-stable scheme with constant and adaptive step-size strategy

Use of partial derivatives to derive a convergent numerical scheme with its error estimates

A new third order convergent numerical solver for continuous dynamical systems

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