Runge-Kutta method continuous representation

The goal is to construct and to evaluate a continuous representation without additional evaluations of the right hand side function /. The construction of a continuous representation is straight forward for integration methods based on a polynomial representation of the solution or its derivative, like Adams or BDF multistep methods or Runge-Kutta methods based on collocation. 

In order to have a continuous representation of the numerical solution without additional evaluations of the rhs-function /, we seek for Runge-Kutta methods of the form... 

This kind of methods allows the use of the same stage values ki for all Xn 9) G [xn,Xn- -i]- If the order of the continuous representation is required to be the order of the global error of the method, then there is such a formulation only for very few Runge-Kutta methods. For the DOPRJ45 method a continuous representation of order 4 exists ... 

For implicit Runge-Kutta methods based on collocation polynomials these polynomials can serve as continuous representation of the solution. Unfortunately for many collocation methods the order of the continuous representation is q < p — 1, so that the requirement at the beginning of this section is not met. 

Theorem 4.5.2 The continuous representation based on a collocation polynomial u of an s-stage implicit Runge-Kutta method is of order s, i.e. 

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