Global Error Estimation with Runge--Kutta Methods

An analysis of global error estimation for Runge—Kuttasolutions of ordinary differential equations is presented. Thebasic technique is that of Zadunaisky in which the global erroris computed from a numerical solution of a neighbouring problemrelated to the main problem by some method of interpolation.It is shown that Runge—Kutta formulae which permit validglobal error estimation using low-degree interpolation can bedeveloped, thus leading to more accurate and computationallyconvenient algorithms than was hitherto expected. Some specialRunge—Kutta processes up to order 4 are presented togetherwith numerical results.