Revised exponentially fitted Runge–Kutta–Nyström methods

D’Ambrosio, Raffaele; Paternoster, Beatrice; Santomauro, Giuseppe

doi:10.1016/j.aml.2013.10.013

Cited by 32 publications

(9 citation statements)

References 25 publications

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“…The structure-preserving approach to SDEs will next be devoted to stochastic Hamiltonian problems [7,8] and the stochastic extension of existing deterministic approaches [32][33][34]. A stochastic version of the non-polynomial fitting for oscillatory problems will also be addressed [35][36][37][38].…”

Section: Discussionmentioning

confidence: 99%

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

Cardone

Conte

D’Ambrosio

et al. 2018

Axioms

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We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of ϑ -methods for Volterra integral equations.

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Section: Discussionmentioning

confidence: 99%

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

Cardone

Conte

D’Ambrosio

et al. 2018

Axioms

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“…where ( ) is the error constant that related to 2 . To determine ( ), following the ideas in [1,2], we evaluate 7and 12…”

Section: Construction Of the New Methodsmentioning

confidence: 99%

“…whose solutions exhibit a pronounced exponential behaviour [1,2]. For the numerical solution of problem (1), classical general-purpose methods such as Runge-Kutta (RK) methods and linear multistep methods (LMMs) can not produce satisfactory results due to the special structure of the problems.…”

Section: Introductionmentioning

confidence: 99%

“…In standard approach to deriving exponentially fitted Runge-Kutta(-Nyström) methods, the effect of the error in the internal stages to the error of the final stage is completely neglected. D' Ambrosio et al revisited the EFRK(N) method with two stages by considering the error in the internal stages in [1,2]. Ixaru [16] introduced A-stable explicit fourth-order Runge-Kutta methods with three stages by evaluating the errors in each internal stage.…”

Section: Introductionmentioning

confidence: 99%

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Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

Yang

Fang

You

et al. 2016

Discrete Dynamics in Nature and Society

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The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods.

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“…(i) Methods with constant coefficients (see [9,7,[10][11][12][13]) and (ii) Methods with coefficients dependent on the frequency of the problem (phase-fitted methods, exponentially-fitted methods and trigonometrically-fitted methods for second-order ODEs) (see [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]). …”

Section: Introductionmentioning

confidence: 99%

An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

Panopoulos

Simos

2015

Journal of Computational and Applied Mathematics

109

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a b s t r a c tOur new linear symmetric semi-embedded predictor-corrector method (SEPCM) presented here is based on the multistep symmetric method of Quinlan and Tremaine (1990), with eight steps and eighth algebraic order and constructed to solve numerically the twodimensional Kepler problem. It can also be used to integrate related IVPs with oscillatory solutions for which the frequency is unknown. Firstly we present a SEPCM (see Panopoulos and Simos, 2013 [36,37]) in pair form. This form has the advantage that reduces the computational expense. From this form we construct a new symmetric eight-step method. The new scheme has constant coefficients and algebraic order ten. We tested the efficiency of our newly developed scheme against some well known methods from the literature. We measure the efficiency of the methods and conclude that the new scheme is the most efficient of all the compared methods and for all the problems solved.

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Revised exponentially fitted Runge–Kutta–Nyström methods

Cited by 32 publications

References 25 publications

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

An eight-step semi-embedded predictor–corrector method for orbital problems and related IVPs with oscillatory solutions for which the frequency is unknown

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