Optimal Method of Runge-Kutta of Order 5

Nonstandard explicit third-order Runge-Kutta method with positivity property

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Based on general theory for positivity, with an explicit third-order Runge-Kutta method (we will refer to it as RK3 method) pos...

Continuous Explicit Runge-kutta Methods of Order 5

A continuous explicit Runge-Kutta (CERK) method provides a continuous approximation to an initial value problem. Such a method may be obtained by appending additional stages to a discrete method, or alternatively by solving the appropriate order conditions directly. Owren and Zennaro have shown for order 5 that the latter approach yields some CERK methods that require fewer derivative evaluatio...

Second-Order Runge-Kutta Approximations in Control Constrained Optimal Control

In this paper, we analyze second-order Runge–Kutta approximations to a nonlinear optimal control problem with control constraints. If the optimal control has a derivative of bounded variation and a coercivity condition holds, we show that for a special class of Runge–Kutta schemes, the error in the discrete approximating control is O(h2) where h is the mesh spacing.

On order 5 symplectic explicit Runge-Kutta Nystro"m methods

Symbolic derivation of Runge-Kutta order conditions

Tree theory, partitions of integer numbers, combinatorial mathematics and computer algebra are the basis for the construction of a powerful and efficient symbolic package for the derivation of Runge Kutta order conditions and principal truncation error terms.