We consider the solution of Hamiltonian dynamical systems by constructing eighth-order explicit symplectic Runge-Kutta-Nystrr om integrators. The application of high-order integrators may be important in areas such as in astronomy. They require large number of function evaluations, which make them computationally expensive and easily susceptible to errors. The integrators developed in this pape...

In this paper, Optimal Homotopy Asymptotic Method (OHAM) is combined with Galerkin method to investigate the MHD Jeffery-Hamel flows in non-parallel walls for strongly nonlinear ordinary differential equations. Results for velocity profiles in divergent and convergent channels for various values of Hartmann and Reynolds numbers are compared with numerical method (the Runge-Kutta method of order...

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties–in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes th...

In this paper three pairs of embedded Runge–Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. ...

We calculated Eq. (4) by using the Runge-Kutta method. The attractors obtained by the simulations are shown in Fig. 3(a). We can confirm the generation of periodic and chaotic attractors. In order to investigate the attractors in van der Pol oscillator under noisy parametric excitation in detail, we consider the Poincaré map. The result of the Poincaré map are shown in Fig. 3(b). We can see tha...

In this work we introduce a new approach to Dynamical Monte Carlo methods to simulate markovian processes. We apply this approach to formulate and study an epidemic generalized SIRS model. The results are in excellent agreement with the fourth order Runge-Kutta method in a region of deterministic solution. Introducing local stochastic interactions, the Runge-Kutta method is no longer applicable...

In this paper the performance of a parallel iterated Runge-Kutta method is compared versus those of the serial fouth order Runge-Kutta and Dormand-Prince methods. It was found that, typically, the runtime for the parallel method is comparable to that of the serial versions, thought it uses considerably more computational resources. A new algorithm is proposed where full parallelization is used ...