A new class of split exponential propagation iterative methods of Runge–Kutta type (sEPIRK) for semilinear systems of ODEs

2014 ◽  
Vol 269 ◽  
pp. 40-60 ◽  
Author(s):  
G. Rainwater ◽  
M. Tokman
Keyword(s):  
Iterative Methods ◽  
Semilinear Systems ◽  
New Class ◽  
Systems Of Odes ◽  
Runge Kutta

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

New Adaptive Exponential Propagation Iterative Methods of Runge--Kutta Type

2012 ◽  
Vol 34 (5) ◽  
pp. A2650-A2669 ◽  
Author(s):  
M. Tokman ◽  
J. Loffeld ◽  
P. Tranquilli
Keyword(s):  
Iterative Methods ◽  
Runge Kutta

scholarly journals Rosenbrock Type Methods for Solving Non-Linear Second-Order in Time Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2225
Author(s):  
Maria Jesus Moreta
Keyword(s):  
Computational Cost ◽  
Intermediate Stage ◽  
Second Order ◽  
First Order ◽  
New Class ◽  
Nyström Methods ◽  
Non Linear ◽  
Second Order In Time ◽  
Linear Problems ◽  
Runge Kutta

In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they can be seen as a generalization of these methods when they are applied to second-order in time problems which have been previously transformed into first-order in time problems. As they also follow the ideas of Runge–Kutta–Nyström methods when solving second-order in time problems, we have called them Rosenbrock–Nyström methods. When solving non-linear problems, Rosenbrock–Nyström methods present less computational cost than implicit Runge–Kutta–Nyström ones, as the non-linear systems which arise at every intermediate stage when Runge–Kutta–Nyström methods are used are replaced with sequences of linear ones.

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