This paper deals with the application of probabilistic time integration methods to semi-explicit partial differential-algebraic equations of parabolic type and its semi-discrete counterparts, namely semi-explicit differential-algebraic equations of index 2. The proposed methods iteratively construct a probability distribution over the solution of deterministic problems, enhancing the information obtained from the numerical simulation. Within this paper, we examine the efficacy of the randomized versions of the implicit Euler method, the midpoint scheme, and exponential integrators of first and second order. By demonstrating the consistency and convergence properties of these solvers, we illustrate their utility in capturing the sensitivity of the solution to numerical errors. Our analysis establishes the theoretical validity of randomized time integration for constrained systems and offers insights into the calibration of probabilistic integrators for practical applications.

翻译：本文研究概率时间积分方法在半显式抛物型偏微分-代数方程及其半离散对应系统（即指标为2的半显式微分-代数方程）中的应用。所提出的方法通过迭代构造确定性问题解的概率分布，增强了数值模拟所获得的信息。本文系统考察了隐式欧拉法、中点格式以及一阶与二阶指数积分器的随机化版本的有效性。通过论证这些求解器的相容性与收敛特性，我们阐明了它们在捕捉解对数值误差敏感性方面的实用价值。本分析为约束系统的随机化时间积分奠定了理论有效性基础，并为概率积分器在实际应用中的参数校准提供了理论依据。