We derive order conditions for the discretization of (unconstrained) optimal control problems, when the scheme for the state equation is of Runge-Kutta type. This problem appears to be essentially the one of checking order conditions for symplectic partitioned Runge-Kutta schemes. We show that the computations using bi-coloured trees are naturally expressed in this case in terms of oriented fre...

Abstract—In this paper zero-dissipative explicit Runge-Kutta method is derived for solving second-order ordinary differential equations with periodical solutions. The phase-lag and dissipation properties for Runge-Kutta (RK) method are also discussed. The new method has algebraic order three with dissipation of order infinity. The numerical results for the new method are compared with existing ...

A staggered Runge-Kutta (staggered RK) scheme is the time integration Runge-Kutta type scheme based on staggered grid, which was proposed by Ghrist and Fornberg and Driscoll in 2000. Afterwords, Vewer presented efficiency of the scheme for linear and semilinear wave equations through numerical experiments. We study stability and convergence properties of this scheme for semilinear wave equation...

Standard Runge-Kutta methods are explicit, one-step, and generally constant step-size numerical integrators for the solution of initial value problems. Such integration schemes of orders 3, 4, and 5 require 3, 4, and 6 function evaluations per time step of integration, respectively. In this paper, we propose a set of simple, explicit, and constant step-size Accerelated-Runge-Kutta methods that ...

The optimum Runge-Kutta method of a particular order is the one whose truncation error is a minimum. Various measures of the size of the truncation error are considered. The optimum method is practically independent of the measure being used. Moreover, among methods of the same order which one might consider using the difference in size of the estimated error is not more than a factor of 2 or 3...

In this paper, we consider the parallel solution of nonstii ordinary diierential equations with two diierent classes of Runge-Kutta (RK) methods providing embedded solutions: classical embedded RK methods and iterated RK methods which were constructed especially for parallel execution. For embedded Runge-Kutta methods, mainly the potential system parallelism is exploited. Iterated RK methods pr...

In this paper, based on the classical Fourth-Order Runge-Kutta method, the modified FourthOrder Runge-Kutta method is presented for solving nonlinear vibration of axially travelling string system, that is to solve time varying and nonlinear differential equations. The classical Fourth-Order Runge-Kutta method can only be used to solve first-order linear differential equations. Its main idea is ...

The cubic spline interpolation method, the Runge–Kutta method, and the Newton–Raphson method are extended to dual versions (developed in the context of dual numbers). This extension allows the calculation of the derivatives of complicated compositions of functions which are not necessarily defined by a closed form expression. The code for the algorithms has been written in Fortran and some exam...