We deal with optimal approximation of solutions of ODEs under local Lipschitz condition and inexact discrete information about the right-hand side functions. We show that the randomized two-stage Runge–Kutta scheme is the optimal method among all randomized algorithms based on standard noisy information. We perform numerical experiments that confirm our theoretical findings. Moreover, for the optimal algorithm we rigorously investigate properties of regions of absolute stability.

中文翻译：

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随机Runge–Kutta方法-不精确信息下的稳定性和收敛性

我们在局部Lipschitz条件下处理ODE解的最佳逼近，并且关于右侧函数的离散信息不精确。我们表明，随机的两阶段Runge-Kutta方案是所有基于标准噪声信息的随机算法中的最佳方法。我们进行了数值实验，证实了我们的理论发现。此外，对于最优算法，我们严格研究了绝对稳定性区域的性质。