The RK5GL3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on a combination of a fifth-order Runge-Kutta method and 3-point Gauss-Legendre quadrature. In this paper we describe an effective local error control algorithm for RK5GL3, which uses local extrapolation with an eighth-order Runge-Kutta method in tandem with RK5GL3, and a ...

System theory for numerical analysis has recently become a focus of research. In this paper we regard dynamics of Newton’s method as a nonlinear feedback system and derive convergence conditions, based on the internal model principle and systems of Lur’e type. We then focus our attention on the analysis of the region of absolute stability of Runge-Kutta type methods. We derive a linear matrix i...

Although Runge–Kutta and partitioned Runge–Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrödinger equation in detail, for which the previously known multisymplectic integrators are fully implicit and based on t...

We consider the scalar autonomous initial value problem as solved by an explicit Runge–Kutta pair of orders 6 and 5. focus on efficient family such pairs, which were studied extensively in previous decades. This comes with 5 coefficients that one is able to select arbitrarily. set, a fitness function, certain measure, evaluated after running couple relevant problems. Thus, we may adjust pair, m...

Although Runge–Kutta and partitioned Runge–Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrödinger equation in detail, for which the previously known multisymplectic integrators are fully implicit and based on t...

‎‎in this article‎, ‎a new method is introduced to give approximate solution to van der pol equation‎. ‎the proposed method is based on the combination of two different methods‎, ‎the spectral adomian decomposition method (sadm) and piecewise method‎, ‎called the piecewise adomian decomposition method (psadm)‎. ‎the numerical results obtained from the proposed method show that this method is an...

in this paper it is attempted to investigate the behavior of an inviscid flow in the meridional plane of an axial flow compressor. for this purpose the 3-d unsteady euler equations in cylindrical coordinate are averaged in tangential direction. therefore, the equations are reduced to a 2-d system. by averaging the tangential component of momentum equation, a blade force will result. axial and r...

This study utilized combination of phase plots,time steps distribution and adaptive time steps Runge-Kutta and f if th order algorithms to investigate a harmonically Duff ing oscillator.The object is to visually compare fourth and f if th order Runge-Kutta algorithms performance as tools for seeking the chaotic solutions of a harmonically excited Duffing oscillator.Though fif th order algorithm...

Recently, the Runge-Kutta methods, obtained via Taylor’s expansion is exist in the literature. In this study, we have derived explicit methods for problems of the form y′ = f(y) including second and third derivatives , by considering available Two-Derivative Runge-Kutta methods (TDRK). The methods use one evaluation of first derivative, one evaluation of second derivative and many evaluations o...