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.

翻译：我们报告了一种新颖的算法，用于在标量、自治、非刚性或弱刚性问题的数值求解中，以逐步（步进式）方式控制全局误差。该算法利用了解析解的泰勒展开余项，需要借助DP853三元组求解辅助问题，进而确定余项值。随后通过淬火过程实现对欧拉方法生成解的控制。我们已实现严格至1e-10的相对全局误差容限。