Stepwise global error control in Euler's method using the DP853 triple and the Taylor remainder term

We report on a novel algorithm for controlling global error in a step-by-step (stepwise) sense, in the numerical solution of a scalar, autonomous, nonstiff or weakly stiff problem. The algorithm exploits the remainder term of a Taylor expansion of the solution. It requires the use of the DP853 triple to solve an auxiliary problem which, in turn, enables the remainder term to be determined. A quenching process then allows the solution generated by Euler’s method to be controlled. We have achieved tolerances on the relative global error as strict as 1e-10.