scholarly journals A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1103
Author(s):  
Felice Iavernaro ◽  
Francesca Mazzia
Keyword(s):  
Fourth Order ◽  
New Method ◽  
Symplectic Method ◽  
Hamiltonian Problems ◽  
First Order ◽  
Runge Kutta Method ◽  
Midpoint Method ◽  
Higher Derivatives ◽  
Runge Kutta ◽  
Order Four

The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.

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.

scholarly journals Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
N. A. Ahmad ◽  
N. Senu ◽  
F. Ismail
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
First Order ◽  
Runge Kutta Method ◽  
Algebraic Order ◽  
Runge Kutta ◽  
The Stability ◽  
Order Four

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

scholarly journals Nine-stage multi-derivative Runge-Kutta method of order 12

2009 ◽  
Vol 86 (100) ◽  
pp. 75-96 ◽  
Author(s):  
Truong Nguyen-Ba ◽  
Vladan Bozic ◽  
Emmanuel Kengne ◽  
Rémi Vaillancourt
Keyword(s):  
Order Conditions ◽  
High Order ◽  
Kutta Method ◽  
True Solution ◽  
First Order ◽  
Cpu Time ◽  
Runge Kutta Method ◽  
Higher Derivatives ◽  
Runge Kutta ◽  
Taylor Methods

A nine-stage multi-derivative Runge-Kutta method of order 12, called HBT(12)9, is constructed for solving nonstiff systems of first-order differential equations of the form y'= f(x, y), y(x0) = y0. The method uses y' and higher derivatives y(2) to y(6) as in Taylor methods and is combined with a 9-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution leads to order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The stepsize is controlled by means of the derivatives y(3) to y(6). The new method has a larger interval of absolute stability than Dormand-Prince's DP(8,7)13M and is superior to DP(8,7)13M and Taylor method of order 12 in solving several problems often used to test high-order ODE solvers on the basis of the number of steps, CPU time, maximum global error of position and energy. Numerical results show the benefits of adding high-order derivatives to Runge-Kutta methods.

scholarly journals A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

2020 ◽  
Vol 17 (1) ◽  
pp. 0166
Author(s):  
Hussain Et al.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods.

scholarly journals Diagonally Implicit Runge-Kutta Fourth Order Four-Stage Method for Linear Ordinary Differential Equations with Minimized Error Norm

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
Nur Izzati Che Jawias ◽  
Fudziah Ismail ◽  
Mohamed Suleiman ◽  
Azmi Jaafar
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
New Method ◽  
Test Problems ◽  
Error Norm ◽  
Linear Ordinary Differential Equations ◽  
Free Parameters ◽  
Runge Kutta ◽  
Order Four

We constructed a new fourth order four-stage diagonally implicit Runge-Kutta (DIRK) method which is specially designed for the integrations of linear ordinary differential equations (LODEs). The method is obtained based on theButcher’s error equations. In the derivation, the error norm is minimized so that the free parameters chosen are obtained from the minimized error norm. Row simplifying assumption is also used so that the number of equations forthe method can be reduced and simplified. A set of test problems are used to validate the method and numerical results show that the new method is more efficient in terms of accuracy compared to the existing method.

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