Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a new way of deriving coefficients of implicit Runge-Kutta methods. This approach based on repeated integrals yields both new and well-known Butcher's tableaux. We discuss the properties of newly derived methods and compare them with standard collocation implicit Runge-Kutta methods in a series of numerical experiments, including the Prothero-Robinson problem.

翻译：在求解常微分方程的数值方法中，龙格-库塔方法具有不可替代的地位。其中，隐式方法特别适用于刚性初值问题的近似求解。本文提出一种推导隐式龙格-库塔方法系数的新途径。这种基于重复积分的方法既能导出新的布彻表，也能得到已知的经典形式。我们通过一系列数值实验（包括Prothero-Robinson问题）讨论了新推导方法的性质，并将其与标准配置型隐式龙格-库塔方法进行了比较。