Exponentially Fitted Two-Derivative Runge-Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems

Chen, Zhaoxia; Li, Juan; Zhang, Ruqiang; You, Xiong

doi:10.1155/2015/689137

Cited by 8 publications

(9 citation statements)

References 28 publications

(34 reference statements)

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“…In this section, we introduce so-called modified two-derivative diagonally implicit Runge-Kutta methods (TDDIRK) for solving (1.1) and derive their order conditions. Our idea was motivated by [14], in which the modified explicit two-derivative Runge-Kutta (TDRK) methods were introduced.…”

Section: Methodsmentioning

confidence: 99%

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Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

Ehigie¹,

Luân²,

Okunuga³

et al. 2020

Preprint

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In this work, we construct and derive a new class of exponentially fitted twoderivative diagonally implicit Runge-Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general format of so-called modified two-derivative diagonally implicit Runge-Kutta methods (TDDIRK) is proposed. Their order conditions up to order six are derived by introducing a set of bi-coloured rooted trees and deriving new elementary weights. Next, we build exponential fitting conditions in order for these modified TDDIRK methods to treat oscillatory solutions, leading to EFTDDIRK methods. In particular, a family of 2-stage fourth-order, a fifth-order, and a 3-stage sixth-order EFTDDIRK schemes are derived. These can be considered as superconvergent methods. The stability and phase-lag analysis of the new methods are also investigated, leading to optimized fourth-order schemes, which turn out to be much more accurate and efficient than their non-optimized versions. Finally, we carry out numerical experiments on some oscillatory test problems. Our numerical results clearly demonstrate the accuracy and efficiency of the newly derived methods when compared with existing implicit Runge-Kutta methods and two-derivative

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Section: Methodsmentioning

confidence: 99%

“…An extensive survey of these methods can be found in [42,54] and the references therein. We also note that explicit EFTDRK methods with optimal phase properties were derived in [17,18,14].…”

Section: Introductionmentioning

confidence: 99%

Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

Ehigie¹,

Luân²,

Okunuga³

et al. 2020

Preprint

Self Cite

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“…Zhang et al [9] proposed a new fifth-order trigonometrically fitted TDRK method for the numerical solution of the radial Schrödinger equation and oscillatory problems. Meanwhile, Fang et al [10] and Chen et al [11] derived two-fourthorder and three practical exponentially fitted TDRK methods, respectively. They compared the new methods with some well-known optimized codes and traditional exponentially fitted RK methods.…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

Ahmad

Senu

Ismail

2017

Mathematical Problems in Engineering

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A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

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“…Zhang et al [20] proposed a new fifth order trigonometrically fitted TDRK method for the numerical solution of the radial Schrödinger equation and oscillatory problems. Meanwhile, Fang et al [12] and Chen et al [5] derived two fourth order and three practical exponentially fitted TDRK methods respectively. They compared the new methods with some well-known optimized codes and traditional exponentially fitted RK methods.…”

Section: Introductionmentioning

confidence: 99%

Trigonometrically-fitted Higher Order Two Derivative Runge-Kutta Method for Solving Orbital and Related Periodical IVPs

Ahmad¹,

Senu²,

Ismail³

2018

HJMS

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In this paper, a trigonometrically-fitted Two Derivative Runge-Kutta method (TFTDRK) of high algebraic order for the numerical integration of first order Initial Value Problems (IVPs) which possesses oscillatory solutions is constructed. Using the trigonometrically-fitted property, a sixth order four stage Two Derivative Runge-Kutta (TDRK) method is designed. The numerical experiments are carried out with the comparison with other existing Runge-Kutta methods (RK) to show the accuracy and efficiency of the derived methods.

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Exponentially Fitted Two-Derivative Runge-Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems

Cited by 8 publications

References 28 publications

Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

Trigonometrically-fitted Higher Order Two Derivative Runge-Kutta Method for Solving Orbital and Related Periodical IVPs

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