Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

1979 ◽  
Vol 33 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Peter Kaps ◽  
Peter Rentrop
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stiff Ordinary Differential Equations ◽  
Stepsize Control ◽  
Runge Kutta ◽  
Order Four

scholarly journals Solving Oscillating Problems Using Modifying Runge-Kutta Methods

2021 ◽  
Vol 34 (4) ◽  
pp. 58-67
Author(s):  
Zainab Khaled Ghazal ◽  
Kasim Abbas Hussain
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lag Phase ◽  
Phase Lag ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Runge Kutta ◽  
Order Four

     This paper develop conventional Runge-Kutta methods of order four and order five to solve ordinary differential equations with oscillating solutions. The new modified Runge-Kutta methods (MRK) contain the invalidation of phase lag, phase lag’s derivatives, and amplification error. Numerical tests from their outcomes show the robustness and competence of the new methods compared to the well-known Runge-Kutta methods in the scientific literature.

RUNGE–KUTTA METHODS FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 15 (08) ◽  
pp. 1203-1251 ◽  
Author(s):  
STEFANO MASET ◽  
LUCIO TORELLI ◽  
ROSSANA VERMIGLIO
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Polynomial Function ◽  
Functional Differential Equations ◽  
Order Conditions ◽  
Polynomial Functions ◽  
Retarded Functional Differential Equations ◽  
Runge Kutta ◽  
Functional Differential ◽  
Order Four

We introduce Runge–Kutta (RK) methods for Retarded Functional Differential Equations (RFDEs). With respect to RK methods (A, b, c) for Ordinary Differential Equations the weights vector b ∈ ℝs and the coefficients matrix A ∈ ℝs×s are replaced by ℝs-valued and ℝs×s-valued polynomial functions b(·) and A(·) respectively. Such methods for RFDEs are different from Continuous RK (CRK) methods where only the weights vector is replaced by a polynomial function. We develop order conditions and construct explicit methods up to the convergence order four.

One Step Cosine–Taylorlike Method For Solving Stiff Equations

2012 ◽  
Author(s):  
Rokiah @ Rozita Ahmad ◽  
Nazeeruddin Yaacob
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Explicit Method ◽  
Stiff Ordinary Differential Equations ◽  
One Step ◽  
Runge Kutta ◽  
New Formula ◽  
Better Than

Makalah ini membincangkan penghasilan kaedah tak tersirat bak Cosine–Taylor untuk menyelesaikan persamaan pembezaan biasa kaku. Perumusannya menghasilkan pengenalan kepada satu rumus baru bagi penyelesaian berangka bagi persamaan pembezaan biasa kaku. Kaedah baru ini memerlukan penghitungan tambahan yakni melakukan beberapa terbitan bagi fungsi yang terlibat. Walau bagaimanapun, keputusan yang diperoleh adalah lebih baik berbanding hasil yang didapati apabila menggunakan kaedah tak tersirat Runge–Kutta peringkat–4 dan kaedah tersirat Adam–Bashfiorth–Moulton (ABM). Perbandingan yang dibuat dengan kaedah bak Sine–Taylor menunjukkan kejituan bagi kedua–dua kaedah adalah hampir setara. Kata kunci: Kaedah tak tersirat; persamaan pembezaan biasa kaku; Runge–Kutta; kaedah tersirat; Adam–Bashforth–Moulton; bak Sine–Taylor This paper discusses the derivation of an explicit Cosine–Taylorlike method for solving stiff ordinary differential equations. The formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations. This new method needs an extra work in order to solve a number of differentiations of the function involved. However, the result produced is better than the results from the explicit classical fourth–order Runge–Kutta (RK4) and the implicit Adam–Bashforth–Moulton (ABM) methods. When compared with the previously derived Sine–Taylorlike method, the accuracy for both methods is almost equivalent. Key words: Explicit method; stiff ordinary differential equations; Runge–Kutta; implicit method; Adam–Bashforth–Moulton; Sine–Taylorlike

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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