A functional fitting Runge-Kutta method with variable coefficients

Ozawa, Kazufumi

doi:10.1007/bf03167357

Cited by 41 publications

(47 citation statements)

References 11 publications

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“…Ozawa [8] shows that if the abscissae ci are taken to satisfy orthogonal condition (28) then the order of accuracy of the FRK method is also being s + v.…”

Section: Indirect Frkn Methodsmentioning

confidence: 99%

A functional fitting Runge-Kutta-Nyström method with variable coefficients

Ozawa

2002

Japan J. Indust. Appl. Math.

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A new type of variable coefficient Runge-Kutta-Nyström methods is proposed for solving the initial value problems of the special form y"(t) = f (t, y(t)). The method is based on the exact integration of some given functions. If the second derivative of the solution is a linear combination of these functions, then the method is exact, and if this is not the case, the algebraic order (order of accuracy) of the method is very important. The algebraic order of the method is investigated by using the power series expansions of the coefficients, which are functions of the stepsize and independent variable t. It is shown that the method has the same algebraic order as those of the direct collocation Runge-Kutta-Nyström method by Van der Houwen et al., when the collocation points are identical with that method. Experimental results which demonstrate the validity of the theoretical analysis are presented.

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“…Ozawa [8] shows that if the abscissae ci are taken to satisfy orthogonal condition (28) then the order of accuracy of the FRK method is also being s + v.…”

Section: Indirect Frkn Methodsmentioning

confidence: 99%

A functional fitting Runge-Kutta-Nyström method with variable coefficients

Ozawa

2002

Japan J. Indust. Appl. Math.

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“…In [8] and [9], Ai = 0 and Fz = A for all i, that is, there exist s unknowns in each of the simultaneous equations, and all the functions Om (t) (m = 1, ..., s) are used to determine these coefficients. Therefore, the resulting method is necessarily a fully implicit one.…”

Section: Functionally Fitted Runge-kutta Methodsmentioning

confidence: 99%

“…Therefore, the resulting method is necessarily a fully implicit one. For this case, Ozawa [8] has shown that the coefficients given by using (2) are unique for all h and t E [0, T], if the Wronskian matrix associated with cp,,,t (t) = 0',, (t) …”

Section: Functionally Fitted Runge-kutta Methodsmentioning

confidence: 99%

“…[2], [7], [10], [11], [13], [14]). To be able to fit Runge-Kutta (-Nystrom) methods to any desired functions, Ozawa [8], [9] has recently developed a technique to construct the RungeKutta (-Nystrom) method that is exact on the linear space of given functions. When the functions are polynomials, the method reduces to the collocation Runge-Kutta methods.…”

Section: Introductionmentioning

confidence: 99%

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A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method

Ozawa

2005

Japan J. Indust. Appl. Math.

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A special class of Runge-Kutta(-Nystrom) methods called functionally fitted (or functional fitting) Runge-Kutta (FRK) methods has recently been proposed by the author. This class of methods is designed to be exact, if the solution of the equation to be solved is an element of the linear space of given functions, which is called the basis functions. The purpose of this article is to develop a functionally fitted Runge-Kutta method that is cheap to implement. The method proposed in this paper is a three-stage explicit singly diagonally implicit Runge-Kutta (ESDIRK) method, which requires one LU decomposition per step. This method is exact if the basis functions are properly chosen, and is moderately accurate even if the choice of the functions is inappropriate, since the method is shown to be of order 4 for general cases. An embedded pair of this type is also developed. Several numerical experiments show the superiority of the methods to conventional ones for the particular case that a suitable set of the basis functions can be found.

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“…In fact, recent studies [7,8,9] have culminated into functionally-fitted methods that allow more general basis functions. Results from these studies have established the existence of such generalized s-stage collocation RK methods for any set of linearly independent basis functions {u i } s i=1 that satisfy the condition that the Wronskian W (u 1 , ..., u s )(h) = 0 for small h > 0.…”

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confidence: 99%

On functionally-fitted Runge–Kutta methods

2006

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Functionally-fitted methods are generalizations of collocation techniques to integrate an equation exactly if its solution is a linear combination of a chosen set of basis functions. When these basis functions are chosen as the power functions, we recover classical algebraic collocation methods. This paper shows that functionally-fitted methods can be derived with less restrictive conditions than previously stated in the literature, and that other related results can be derived in a much more elegant way. The novelty in our approach is to fully retain the collocation framework without reverting back into derivations based on cumbersome Taylor series expansions. (2000): 65L05, 65L06, 65L20, 65L60. AMS subject classification

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A functional fitting Runge-Kutta method with variable coefficients

Cited by 41 publications

References 11 publications

A functional fitting Runge-Kutta-Nyström method with variable coefficients

A functional fitting Runge-Kutta-Nyström method with variable coefficients

A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method

On functionally-fitted Runge–Kutta methods

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