Interpolants for Runge-Kutta pairs of order four and five

Computing ◽  
1990 ◽  
Vol 45 (4) ◽  
pp. 383-388
Author(s):  
M. Calvé ◽  
R. Vaillancourt
Keyword(s):  
Runge Kutta ◽  
Order Four

scholarly journals Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4)=f(x,u,u′,u′′)

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 246 ◽  
Author(s):  
Nizam Ghawadri ◽  
Norazak Senu ◽  
Firas Fawzi ◽  
Fudziah Ismail ◽  
Zarina Ibrahim
Keyword(s):  
Real Life ◽  
Order Ordinary Differential Equation ◽  
New Techniques ◽  
First Order ◽  
New Methods ◽  
Life Problems ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Order Four ◽  
Primary Contribution

The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure u ( 4 ) = f ( x , u , u ′ , u ′ ′ ) and denoted as RKTF method. We presented the associated B-series and quad-colored tree theory with the aim of deriving the prerequisites of the said order. Depending on the order conditions, the method with algebraic order four with a three-stage and order five with a four-stage denoted as RKTF4 and RKTF5 are discussed, respectively. Numerical outcomes are offered to interpret the accuracy and efficacy of the new techniques via comparisons with various currently available RK techniques after converting the problems into a system of first-order ODE systems. Application of the new methods in real-life problems in ship dynamics is discussed.

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 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.

scholarly journals On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 924 ◽  
Author(s):  
Khai Chien Lee ◽  
Norazak Senu ◽  
Ali Ahmadian ◽  
Siti Nur Iqmal Ibrahim
Keyword(s):  
Thin Film Flow ◽  
Third Order ◽  
New Methods ◽  
Algebraic Order ◽  
Fourth Derivative ◽  
Three Stages ◽  
Runge Kutta ◽  
Two Stages ◽  
Order Four ◽  
Third Derivative

A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form u ‴ ( x ) = f ( x , u ( x ) ) . In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments.

EXPONENTIALLY FITTED TWO-DERIVATIVE RUNGE–KUTTA METHODS FOR THE SCHRÖDINGER EQUATION

2013 ◽  
Vol 24 (10) ◽  
pp. 1350073 ◽  
Author(s):  
YONGLEI FANG ◽  
XIONG YOU ◽  
QINGHE MING
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
High Efficiency ◽  
New Methods ◽  
Asymptotic Expressions ◽  
Radial Schrödinger Equation ◽  
Algebraic Order ◽  
Exponentially Fitted ◽  
Runge Kutta ◽  
Order Four

Two exponentially fitted two-derivative Runge–Kutta (EFTDRK) methods of algebraic order four are derived. The asymptotic expressions of the local errors for large energies are obtained. The numerical results in the integration of the radial Schrödinger equation with the Woods–Saxon potential show the high efficiency of our new methods compared to some famous optimized codes in the literature.

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