A Survey of Interval Runge–Kutta and Multistep Methods for Solving the Initial Value Problem

Gajda, Karol; Jankowska, Małgorzata A.; Marciniak, Andrzej; Szyszka, Barbara

doi:10.1007/978-3-540-68111-3_144

Cited by 22 publications

(15 citation statements)

References 19 publications

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“…. , n − 1) one can obtain by applying an interval one-step method, for example an interval method of Runge-Kutta type (see, e.g., [7,8,20,27,31,43]) or interval methods based on Taylor series (see, e.g., [2,3,5,15,21,39]). Let us note that in (10), we cannot write γ * n + γ * * n n instead of γ * n n + γ * * n n , because in general γ * n + γ * * n may be different from γ * n + γ * * n .…”

Section: And Where (T Y ) Denotes An Interval Extension Of F (N) (Tmentioning

confidence: 99%

“….9] and let us take additional starting intervals presented in Table 5 (these intervals have been obtained by an interval version of conventional fourth order Runge-Kutta method [7,20,27,43] with h = 0.0001). In Table 6, we present the results obtained at t = 1 by the methods N4 and M4.…”

Section: Example 1 Firstly Let Us Consider the Commonly Used Test Prmentioning

confidence: 99%

“…h = 0.0001, and taking additional starting intervals given in Table 10 (these intervals have been obtained by an interval version of conventional fourth order Runge-Kutta method [7,20,27,43] with h = 0.00001), we have obtained by the M4 method the results presented in Table 11. If the integration interval is sufficiently small, the results obtained by the low (fourth) order N4 and M4 methods can be compared with the results obtained by methods based on high-order Tylor series.…”

Section: Example 1 Firstly Let Us Consider the Commonly Used Test Prmentioning

confidence: 99%

“…The first one was described by R. E. Moore in 1965 [33][34][35]. There are also known interval methods based on high-order Taylor series (see, e.g., [2,3,5,15,21,39,41]), explicit and implicit Runge-Kutta methods [7,8,20,24,27,31,43], explicit and implicit multistep methods [17-19, 25-27, 30, 43]. The last ones concern interval methods based on conventional methods of Adams-Bashforth, Adams-Moulton, Nyström, and Milne-Simpson types.…”

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confidence: 99%

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Interval versions of Milne’s multistep methods

Marciniak

Jankowska

2017

Numer Algor

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The paper presents explicit interval multistep methods of Milne type, which may be considered as alternative methods to other known explicit interval multistep methods (of Adams-Bashforth and Nyström). It is proved that enclosures of solutions (in the form of intervals) obtained by these methods contain the exact solutions of the initial value problem. Numerical examples show that the widths of intervals obtained by proposed methods are smaller than those obtained by explicit interval multistep methods known so far.

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Section: And Where (T Y ) Denotes An Interval Extension Of F (N) (Tmentioning

confidence: 99%

Section: Example 1 Firstly Let Us Consider the Commonly Used Test Prmentioning

confidence: 99%

Section: Example 1 Firstly Let Us Consider the Commonly Used Test Prmentioning

confidence: 99%

mentioning

confidence: 99%

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Interval versions of Milne’s multistep methods

Marciniak

Jankowska

2017

Numer Algor

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“…An interval method for ordinary differential equations using interval arithmetic was described first by R. E. Moore in 1965 [32,33]. There are also interval methods based on explicit Runge-Kutta methods [21,28,41] and implicit ones [10,11,25,28,31]. In [41], Yu.…”

Section: Introductionmentioning

confidence: 99%

On interval predictor-corrector methods

2016

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One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of RungeKutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.

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Applications of Interval B&BT Methods

Kubica

2019

Interval Methods for Solving Nonlinear Constraint Satisfaction, Optimization and Similar Problems

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A Survey of Interval Runge–Kutta and Multistep Methods for Solving the Initial Value Problem

Cited by 22 publications

References 19 publications

Interval versions of Milne’s multistep methods

Interval versions of Milne’s multistep methods

On interval predictor-corrector methods

Applications of Interval B&BT Methods

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