An interval version of the Kuntzmann-Butcher method for solving the initial value problem

Abstract

The Kutzmann-Butcher method is the unique implicit four-stage Runge-Kutta method of order 8. In many problems in ordinary differential equations this method realized in floating-point arithmetic gives quite good approximations to the exact solutions, but the results obtained do not contain any information on rounding errors, representation errors and the error of the method. Thus, we describe an interval version of this method, which realized in floating-point interval arithmetic gives approximations (enclosures in the form of interval) containing all these errors. The described method can also include data uncertainties in the intervals obtained.