On functionally-fitted Runge–Kutta methods

Hoang, N. S.; Sidje, Roger B.; Công, Nguyễn Hữu

doi:10.1007/s10543-006-0092-x

Cited by 19 publications

(22 citation statements)

References 14 publications

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“…We briefly summarize the results given in [6] regarding this aspect. Definition 2.2 (Collocation condition).…”

Section: The Collocation Conditionmentioning

confidence: 97%

“…As stated in [6], the practical implication here is that the coefficients of an FRK method based on basis functions that satisfy the collocation condition are uniquely determined almost everywhere on the integration domain. Additionally, it is shown in [8,6] that an s-stage FRK method has a stage order s and a step order at least s and at most 2s.…”

Section: The Collocation Conditionmentioning

confidence: 99%

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On the stability of functionally fitted Runge–Kutta methods

Hoang

Sidje

2008

Bit Numer Math

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Classical collocation RK methods are polynomially fitted in the sense that they integrate an ODE problem exactly if its solution is an algebraic polynomial up to some degree. Functionally fitted RK (FRK) methods are collocation techniques that generalize this principle to solve an ODE problem exactly if its solution is a linear combination of a chosen set of arbitrary basis functions. Given for example a periodic or oscillatory ODE problem with a known frequency, it might be advantageous to tune a trigonometric FRK method targeted at such a problem. However, FRK methods lead to variable coefficients that depend on the parameters of the problem, the time, the stepsize, and the basis functions in a non-trivial manner that inhibits any in-depth analysis of the behavior of the methods in general. We present the class of so-called separable basis functions and show how to characterize the stability function of the methods in this particular class. We illustrate this explicitly with an example and we provide further insight for separable methods with symmetric collocation points. (2000): 65L05, 65L06, 65L20, 65L60. AMS subject classification

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“…We briefly summarize the results given in [6] regarding this aspect. Definition 2.2 (Collocation condition).…”

Section: The Collocation Conditionmentioning

confidence: 97%

Section: The Collocation Conditionmentioning

confidence: 99%

On the stability of functionally fitted Runge–Kutta methods

Hoang

Sidje

2008

Bit Numer Math

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“…In [5] is introduced numerical integration of differential algebraic systems. The use of Runge -Kutta method to solve non-linear system of differential equations is presented in [3]. When time increase (or other parameter used as a step) is variable, it is appropriate to use modified Runge -Kutta -Nyström method, which is modified for variable step of numerical integration.…”

Section: Introductionmentioning

confidence: 99%

Application of Numerical Integration in Technical Practice

Rédl¹,

Váliková²,

Páleš³

et al. 2015

MERAA

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In this contribution we present the methodology of the solution of ordinary differential equation by the Runge -Kutta numerical method of fourth order. Analysed function is continuous and it has derivatives at every point. Records of the centre of the gravity velocities of agricultural technological vehicle are function values of the derivatives. Algorithm of numerical integration was implemented by the help of programming language C# in MS Visual Studio Pro 2010 developing environment. Trajectory of the centre of the gravity movement in three-dimensional space is the result of the listed algorithm.

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“…These methods are developed to integrate an ODE exactly if the solution of the ODE is a linear combination of some certain basis functions (see, e.g., [2], [7], [8], [12], [13], [14], [17]). For example, trigonometric methods have been developed to solve periodic or nearly periodic problems (see, e.g., [7], [11], [17]).…”

Section: Introductionmentioning

confidence: 99%

“…For example, trigonometric methods have been developed to solve periodic or nearly periodic problems (see, e.g., [7], [11], [17]). Numerical experiments have shown that trigonometrically-fitted methods are superior to classical Runge-Kutta methods for solving ODEs whose solutions are periodic or nearly periodic functions with known frequencies (see, e.g., [8], [11], [12], [13], [14], [16], [17]). …”

Section: Introductionmentioning

confidence: 99%

Functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström methods

Hoang

2015

Applied Numerical Mathematics

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A general class of functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström (FEPTRKN) methods for solving second-order initial value problems has been studied. These methods can be considered generalized explicit pseudo two-step Runge-Kutta-Nyström (EPTRKN) methods. We proved that an s-stage FEPTRKN method has step order p = s and stage order r = s for any set of distinct collocation parameters (ci). Supperconvergence for the accuracy orders of these methods can be obtained if the collocation parameters (ci) s i=1 satisfy some orthogonality conditions. We proved that an s-stage FEPTRKN method can attain accuracy order p = s + 3. Numerical experiments have shown that the new FEPTRKN methods work better than do EPTRKN methods on problems whose solutions can be well approximated by the functions in bases on which these FEPTRKN methods are developed.

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On functionally-fitted Runge–Kutta methods

Cited by 19 publications

References 14 publications

On the stability of functionally fitted Runge–Kutta methods

On the stability of functionally fitted Runge–Kutta methods

Application of Numerical Integration in Technical Practice

Functionally-fitted explicit pseudo two-step Runge–Kutta–Nyström methods

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