On bounding the errors in three Runge-Kutta type formulations

Abstract

Total error bounds are established for three of the Runge—Kutta type numerical integration formulas. These are (a) the Heun third-order formula, (b) the Kutta—Simpson three-eighths formula, and (c) the Kutta—Simpson one-third formula. For (a) and (b), it is established that when analyzing for equations of the sort y′(x) = ƒ(x,y), the total error at any step i is of the form | E i | ≤ E(e ihM − 1)/(e hm − 1) where h is the step size and E and M are certain constants. It is found that a system of two equations, y′(x) = ƒ(x, y, z) and z′( g) = ( g)( x, y, z), have a joint error given by max (| E i |, | D i |) ≤ E(e 2 ihM − 1)/(e 2 hM − 1) when analyzed by method (c). In addition to error bounds, conditions under which the ith step result is valid are derived for each procedure.