The RK1GL2X3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on the RK1GL2 method which, in turn, is a particular case of the general RKrGLm method. The RK1GL2X3 method is a fourth-order method, even though its underlying Runge-Kutta method RK1 is the first-order Euler method, and hence, RK1GL2X3 is considerably more efficient tha...

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...

We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series) which we call aromatic B-series. We obtain an explicit description of aromatic B-series in terms of elementary differentials associated to aromatic trees, which are directed graphs generalizing trees. We also define a new...

The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obta...

When solving certain evolution type PDE such as the Schrödinger equation, the Interaction Picture method is a valuable alternative to Split-Step methods. The Interaction Picture method has good computational features when used together with the standard 4th order Runge-Kutta scheme (giving rise to the RK4-IP method). In this paper we present an embedded Runge-Kutta scheme with orders 3 and 4wit...

In this work a systematic procedure is implemented in order to minimise the computational cost of the Runge–Kutta–Munthe-Kaas (RKMK) class of Lie-group solvers. The process consists of the application of a linear transformation to the stages of the method and the analysis of a graded free Lie algebra to reduce the number of commutators involved. We consider here RKMK integration methods up to o...

A backward analysis of integration methods, whose numerical solution is a P-series, is presented. Such methods include Runge-Kutta methods, partitioned Runge-Kutta methods and Nystrr om methods. It is shown that the numerical solution can formally be interpreted as the exact solution of a perturbed diierential system whose right-hand side is again a P-series. The main result of this article is ...

Usually the straightforward generalization of explicit Runge{Kutta methods for ordinary diierential equations to half-explicit methods for diierential-algebraic systems of index 2 results in methods of order q 2 ((8]). The construction of higher order methods is simpliied substantially by a slight modiication of the method combined with an improved strategy for the computation of the algebraic ...

In this paper we consider the stability of variable step-size Runge-Kutta methods approximating bounded, stable, and time-dependent solutions of ordinary differential equation initial value problems. We use Lyapunov exponent theory to determine conditions on the maximum allowable step-size that guarantees the numerical solution of an asymptotically decaying time-dependent linear problem also de...

The equations governing the flow of an electrically conducting, incompressible viscous fluid over an infinite flat plate in the presence of a magnetic field are investigated using the homotopy perturbation method (HPM) with Padé approximants (PA) and 4 order Runge–Kutta method (4RKM). Approximate analytical and numerical solutions for the velocity field and heat transfer are obtained and compar...