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Runge - Kutta Methods: Analysis and Implementation

Thesis Info

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Author

Iqbal, Junaid

Program

PhD

Institute

University of Engineering and Technology

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/12448/1/junaid%20ahmad%20maths%202019%20uet%20lhr%20prr.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727129680

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In this thesis, we have developed new numerical methods in Runge-Kutta family for numerical solution of ordinary di erential equations. We have extended the idea of e ective order to Runge-Kutta Nystr om methods for numerical approximation of second order ordinary di erential equations. The composition of Runge-Kutta Nystr om methods, the pruning of associated Nystr om trees, and conditions for e ective order Runge-Kutta Nystr om methods up to orderve are presented. Also, partitioned Runge-Kutta methods of e ective order 4 with 3 stages are constructed. The most obvious feature of these methods is e ciency in terms of implementation cost. The numerical results verify that the asymptotic error behavior of the e ective order 4 partitioned Runge- Kutta methods with 3 stages is similar to that of classical order 4 method which necessarily require 4 stages. Moreover, it is evident from the numerical results that e ective order methods are more e cient than their classical order counterpart. Lastly, a family of explicit symplectic partitioned Runge-Kutta methods are derived with e ective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more e cient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian system con rming the e ciency of the approach is also provided with good energy conservation.
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