Analysis of starting approximations for implicit Runge-Kutta methods applied to ODEs based on the reverse method

Abstract

We consider the application of s-stage implicit Runge-Kutta methods to ordinary differential equations (ODEs). We consider starting approximations based on values from the previous step to obtain an accurate initial guess for the internal stages of the current step. To simplify the analysis of those starting approximations we compare the expansions of the starting approximation and of the exact value of the internal stages at the initial value $x_n$ of the current step and not at the initial value $x_{n-1}$ of the previous step. In particular, for the starting approximation we make use of the expansion of the reverse IRK method from the initial value $x_n$ of the current step with a negative step size. This simplifies considerably the expression of the order conditions. As a consequence it allows us to give more general and precise statements about the existence and uniqueness of a starting approximation of a given order for IRK methods satisfying the simplifying assumptions $B\left(p\right)$ and $C\left(q\right)$. In particular we show under certain assumptions the nonexistence of starting approximations of order $s+1$ for the type of starting approximations considered.