Global Error Estimation with Runge—Kutta Methods

An analysis of global error estimation for Runge—Kutta solutions of ordinary differential equations is presented. The basic technique is that of Zadunaisky in which the global error is computed from a numerical solution of a neighbouring problem related to the main problem by some method of interpolation. It is shown that Runge—Kutta formulae which permit valid global error estimation using low-degree interpolation can be developed, thus leading to more accurate and computationally convenient algorithms than was hitherto expected. Some special Runge—Kutta processes up to order 4 are presented together with numerical results.