In this paper, we introduce a randomized version of the backward Euler method that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carathéodory-type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of 0.5 in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter. These results are then extended to the approximation of infinite-dimensional evolution equations under monotonicity and Lipschitz conditions. Here, we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.

中文翻译：

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时间不规则系数非线性发展方程的随机向后欧拉方法

在本文中，我们介绍了后向Euler方法的随机版本，该方法适用于具有时间不规则系数的刚性常微分方程和非线性发展方程。在有限维情况下，我们考虑满足单侧Lipschitz条件的Carathéodory型函数。在研究了随机方案的适定性和稳定性之后，我们证明了均方根范数中以0.5的比率收敛到精确解的前提是仅假设系数函数相对于时间是平方可积的参数。然后将这些结果推广到单调性和Lipschitz条件下的无穷维发展方程的逼近。这里，我们考虑将随机后向Euler方案与Galerkin有限元方法结合起来。我们获得与精确解的规律性相对应的误差估计。通过几个数值实验也说明了随机方案的实用性。