Local accuracy and error bounds of the improved Runge-Kutta numerical methods

• Autorzy: Qureshi S. , Memon Z. , Shaikh A. A.

Identyfikatory

DOI

10.17512/jamcm.2018.4.08

• Języki publikacji: EN

Abstrakty

EN

In this paper, explicit Improved Runge-Kutta (IRK) methods with two, three and four stages have been analyzed in detail to derive the error estimates inherent in them whereas their convergence, order of local accuracy, stability and arithmetic complexity have been proved in the relevant literature. Using single and multivariate Taylor series expansion for a mathematical function of one and two variables respectively, slopes involved in the IRK methods have been expanded in order to obtain the general expression for the leading or principal term in the local truncation error of the methods. In addition to this, principal error functions of the methods have also been derived using the idea of Lotkin bounds which consequently gave rise to the error estimates for the IRK methods. Later, these error estimates were compared with error estimates of the two, three, and four-stage standard explicit Runge-Kutta (RK) methods to show the better performance of the IRK methods in terms of the error bounds on the constant step-size h used for solving the initial value problems in ordinary differential equations. Finally, a couple of initial value problems have been tested to determine the maximum absolute global errors, absolute errors at the final nodal point of the integration interval and the CPU times (seconds) for all the methods under consideration to get a better idea of how the methods behave in a particular situation especially when it comes to analyzing the error terms.

Słowa kluczowe

EN

error estimate; remainder term; principal error function; truncation error; Lotkin bound

PL

granice błędu; metoda numeryczna Runge-Kutty; błąd obcięcia; granica Lotkina; funkcja błędu; szacowanie błędu

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2018
• Tom: Vol. 17, nr 4
• Strony: 73--84
• Opis fizyczny: Bibliogr. 21 poz. tab.

Twórcy

autor

Qureshi S.

• Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology Jamshoro, Pakistan

autor

Memon Z.

• Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology Jamshoro, Pakistan

autor

Shaikh A. A.

• Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology Jamshoro, Pakistan

Bibliografia

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Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).