A parallel algorithm for the estimation of the global error in Runge-Kutta methods

The object of this work is the estimate of the global error in the numerical solution of the IVP for a system of ODE's. Given a Runge-Kutta formula of order q, which yields an approximation y(n) to the true value y(x(n)), a general, parallel method is presented, that provides a second value y(n)* of order q + 2; the global error e(n) = y(n) - y(x(n)) is then estimated by the difference y(n) - y(n)*. The numerical tests reported, show the very good performance of the procedure proposed. A comparison with the code GEM90 is also appended.