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 ...

We consider a very general class of Runge-Kutta methods for the numerical solution of Volterra integral equations of the second kind, which includes as special cases all the more important methods which have been considered in the literature. The main purpose of this paper is to define and prove the existence of the Natural Continuous Extensions (NCE's) of Runge-Kutta methods, i.e., piecewise p...

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...

Abstract: In this paper, the fifth order Improved Runge-Kutta method (IRK5) that uses just five function evaluations per step is developed. The method proposed here are derived with only five stages which results in lower number of function evaluations. Therefore, IRK5 has a lower computational cost than the classical fifth order Runge-Kutta method (RK5). Here, the order conditions of the metho...

We consider implicit-explicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stabilitypreserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge Kutta (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space...

In this paper we consider the numerical solution of 2D systems of certain types of taxis-diiusion-reaction equations from mathematical biology. By spatial discretization these PDE systems are approximated by systems of positive, nonlinear ODEs (Method of Lines). The aim of this paper is to examine the numerical integration of these ODE systems for low to moderate accuracy by means of splitting ...