A family of A-stable Runge Kutta collocation methods of higher order for initial-value problems

2007 ◽  
Vol 27 (4) ◽  
pp. 798-817 ◽  
Author(s):  
J. Vigo-Aguiar ◽  
H. Ramos
Keyword(s):  
Initial Value Problems ◽  
Higher Order ◽  
Collocation Methods ◽  
Initial Value ◽  
Runge Kutta

Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems

2017 ◽  
Vol 8 (1-2) ◽  
pp. 118
Author(s):  
G. D. Yakubu ◽  
G. M. Kumleng ◽  
S. Markus
Keyword(s):  
Initial Value Problems ◽  
Absolute Stability ◽  
Collocation Methods ◽  
Test Problems ◽  
Second Derivative ◽  
Stiff Systems ◽  
Initial Value ◽  
Lipschitz Constants ◽  
Runge Kutta ◽  
Second Derivative Methods

Second derivative Runge-Kutta collocation methods for the numerical solution of sti system of rst order initial value problems in ordinary dierential equations are derived and studied. The inclusion of the second derivative terms enabled us to derive a set of methods which are convergent with large regions of absolute stability. Although the implementation of the methods remains iterative in a precisely dened way, the advantage gained makes them suitable for solving sti system of equations with large Lipschitz constants. The derived methods are illustrated by the applications to some test problems of sti system found in the literature and the numerical results obtained conrm the potential of the second derivative methods.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

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