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...

Citation A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and We compare the three main types of high-order one-step initial value solvers: extrapolation, spectral deferred correction, and embedded Runge–Kutta pairs. We consider orders four through twelve, including both serial and parallel implementations. We cast extrapolation and defer...

New software based on explicit Runge-Kutta formulas have been developed to replace well-established, widely-used codes written by the authors (RKF45 and its successors in the SLATEC Library and the NAG Fortran 77 Library Runge-Kutta codes). The new software has greater functionality than its predecessors. Also, it is more eecient, more robust and better documented.

A new selection is made of the most practical of the many explicit Runge-Kutta formulas of order 4 which have been proposed. A new formula is considered, formulas are modified to improve their quality and efficiency in agreement with improved understanding of the issues, and formulas are derived which permit interpolation. It is possible to do a lot better than the pair of Fehlberg currently re...

Taking into account that many problems of natural sciences and engineering are reduced to solving initial-value problem for ordinary differential equations, beginning from Newton, the scientists investigate approximate solution of ordinary differential equations. There are papers of different authors devoted to the solution of initial value problem for ODE. The Euler’s known method that was dev...

Generalized Runge-Kutta Processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a Jacobian at every step. However, it is possible to derive processes which are Internally S-stable when an accurate Jacobian is used but still remain consistent and highly stable if an approximate Jacobian is used. It is shown that these processes require at least as...