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. In particular, we observe higher accuracy and higher experimental order of convergence of some newly derived methods.

翻译：龙格-库塔方法在求解常微分方程的数值方法中具有不可替代的地位。特别是隐式方法适用于逼近刚性初值问题的解。我们提出了一种推导隐式龙格-库塔方法系数的新途径。该方法基于重复积分，既能得到新的Butcher表，也能得到已知的Butcher表。我们讨论了新推导方法的性质，并通过一系列数值实验将其与标准配置隐式龙格-库塔方法进行了比较。特别地，我们观察到部分新推导方法具有更高的精度和更高的实验收敛阶。