In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

中文翻译：

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提高Runge-Kutta型搭配方法求解ODE的准确性

在本文中，提出了一种用于普通微分方程（ODE）数值积分的新型Runge–Kutta型配置方法。它的推导基于微分方程的积分形式。该方法可以提高已建立的并置Runge-Kutta方法的准确性，同时保留相同数量的阶段。我们证明，通过所提出的方法，可以推导出高斯-勒根德尔和洛巴托IIIA方法，并且对于相同数量的方法系数，它们的准确性可以得到提高。我们以类似于Butcher tableaus的表的形式表示方法。在文献中对一些著名的刚性，振荡和非线性ODE进行了新方法性能的研究。