A-stable parallel block methods for ordinary and integro-differential equations

Sommeijer, B. P.; Couzy, W.; Houwen, P. J. van der

doi:10.1016/0168-9274(92)90021-5

Cited by 28 publications

(8 citation statements)

References 6 publications

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“…(1.3') are straightforwardly derived (cf. [22]) and are given by (2.2) Cj=O, j=l,···,r, C1 :=a+Ae-c, Cj:=jAcj-I_cJ, j=2,3,···, where cj denotes the vector with components ( c;)j. Thus, to achieve stage order r for a given block point vector c, we have to solve rk linear equations in k 2 + k unknowns, so that the maximal stage order equals k + 1.…”

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confidence: 99%

Iterated Runge–Kutta Methods on Parallel Computers

Houwen¹,

Sommeijer²

1991

SIAM J. Sci. and Stat. Comput.

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Abstract. This paper examines diagonally implicit iteration methods for solving implicit Runge-Kutta methods with high stage order on parallel computers. These iteration methods are such that after a finite number of m iterations, the iterated Runge-Kutta method belongs to the class of diagonally implicit Runge-Kutta methods (DIRK methods) using mk implicit stages where k is the number of stages of the generating implicit Runge-Kutta method (corrector method). However, a large number of the stages of this DIRK method can be computed in parallel, so that the number of stages that have to be computed sequentially is only m. The iteration parameters of the method are tuned in such a way that fast convergence to the stability characteristics of the corrector method is achieved. By means of numerical experiments it is also shown that the solution produced by the resulting iteration method converges rapidly to the corrector solution so that both stability and accuracy characteristics are comparable with those of the corrector. This implies that the reduced accuracy often shown when integrating stiff problems by means of DIRK methods already available in the literature (which is caused by a low stage order) is not shown by the DIRK methods developed in this paper, provided that the corrector method has a sufficiently high stage order.

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confidence: 99%

Iterated Runge–Kutta Methods on Parallel Computers

Houwen¹,

Sommeijer²

1991

SIAM J. Sci. and Stat. Comput.

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“…One-and multi-block methods advance the numerical solution by a block of more than one new solution values at a time and enjoy high accuracy and good stability (see, e.g., Andria, Byrne and Hill [1], Bond and Cash [2], Chartier [9], Chu and Hamilton [10], Iavernaro and Mazzia [20], Lu [26], Sommeijer, Couzy and van der Houwen [38], Tian, Shan and Kuang [40], Tian, Yu and Jin [41], Watanabe [44], Xie and Tian [48] and Zhou [50]).…”

Section: Jingwen Wu Jintao Hu and Hongjiong Tianmentioning

confidence: 99%

Functionally-fitted block <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>-methods for ordinary differential equations

Wu¹,

Hu²,

Tian³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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We propose a new family of functionally-fitted block θ-methods for numerically solving ordinary differential equations which integrates a chosen set of linearly independent functions exactly. The advantage of such variable coefficient methods is that the basis functions can be chosen to exploit specific properties of the problem that may be known in advance. The basic theory for the proposed methods is established. First, we derive a sufficient condition to ensure the existence of the functionally-fitted block θ-methods, and discuss the independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of r-dimensional functionallyfitted block θ-method is at least r for ordinary differential equations. Numerical experiments are conducted to illustrate the efficiency of the functionally-fitted block θ-methods. 1. Introduction. Consider the following ordinary differential equations (ODEs) y (t) = f (t, y(t)), t ∈ [t 0 , T ], y(t 0) = y 0. (1) Suppose that f : [t 0 , T ] × R → R is continuous and satisfies a Lipschitz condition |f (t, y) − f (t, z)| ≤ L|y − z|, ∀t ∈ [t 0 , T ], y, z ∈ R such that there exists a unique solution. We shall assume that y(t) has continuous derivatives on [t 0 , T ] of any order needed. Numerous classical numerical methods including Runge-Kutta methods, multistep methods and block methods have

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“…BMMs can be considered as a set of simultaneously applied LMMs to obtain several numerical approximations within each integration step (Sommeijer et al [1]). For excellence surveys and various perspectives of BMMs, see, for example, Sommeijer et al [1], Watanabe [2], Ibrahim et al [3][4], Chollom et al [5], Majid et al [6][7], Mehrkanoon et al [8], Akinfenwa et al [9], Ehigie et al [10], Ibijola et al [11] and Majid and Suleiman [12].…”

Section: Introductionmentioning

confidence: 99%

“…For excellence surveys and various perspectives of BMMs, see, for example, Sommeijer et al [1], Watanabe [2], Ibrahim et al [3][4], Chollom et al [5], Majid et al [6][7], Mehrkanoon et al [8], Akinfenwa et al [9], Ehigie et al [10], Ibijola et al [11] and Majid and Suleiman [12]. Implicit BMMs were introduced mainly to improve the order of consistencies and stability requirements suffered by most explicit BMMs.…”

Section: Introductionmentioning

confidence: 99%

“…Firstly, BMMs can easily be modified and extended to solve higher order initial value problems directly, as reported in Majid et al [6][7], Ehigie et al [10], Badmus and Yahaya [13] and Olabode [14]. Secondly, BMMs can easily be implemented on a parallel machine, as reported in Sommeijer et al [1], Mehrkanoon et al [8] and Chartier [15]. Thus, the potential of BMMs is obvious regardless of their stability drawbacks.…”

Section: Introductionmentioning

confidence: 99%

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Block multistep methods based on rational approximants

Ying¹,

Omar²,

Mansor³

2014

AIP Conference Proceedings

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Variable step direct block multistep method for general second order ODEs AIP Conf.A rational approximation for the parabolic equation method A parabolic wave equation based on a rational-cubic approximation J. Acoust. Soc. Am. 87, 619 (1990); 10.1121/1.398930 N−variable rational approximants and method of moments Abstract. In this study, the concept of block multistep methods based on rational approximants is introduced for the numerical solution of first order initial value problems. These numerical methods are also called rational block multistep methods. The main reason to consider block multistep methods in rational setting, is to improve the numerical accuracy and absolute stability property of existing block multistep methods that are based on polynomial approximants. For this pilot study, a 2-point explicit rational block multistep method is developed. Local truncation error and stability analysis for this new method are included as well. Numerical experimentations and results using some test problems are presented. Numerical results are satisfying in terms of numerical accuracy. Finally, future issues on the developments of rational block multistep methods are discussed.

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A-stable parallel block methods for ordinary and integro-differential equations

Cited by 28 publications

References 6 publications

Iterated Runge–Kutta Methods on Parallel Computers

Iterated Runge–Kutta Methods on Parallel Computers

Functionally-fitted block <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula>-methods for ordinary differential equations

Block multistep methods based on rational approximants

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