Two combined numerical methods for solving time-varying semilinear differential-algebraic equations (DAEs) are obtained. The convergence and correctness of the methods are proved. When constructing the methods, time-varying spectral projectors which can be found numerically are used. This enables to numerically solve the DAE in the original form without additional analytical transformations. To improve the accuracy of the second method, recalculation is used. The developed methods are applicable to the DAEs with the continuous nonlinear part which may not be differentiable in time, and the restrictions of the type of the global Lipschitz condition are not used in the presented theorems on the DAE global solvability and the convergence of the methods. This extends the scope of methods. The fulfillment of the conditions of the global solvability theorem ensures the existence of a unique exact solution on any given time interval, which enables to seek an approximate solution also on any time interval. Numerical examples illustrating the capabilities of the methods and their effectiveness in various situations are provided. To demonstrate this, mathematical models of the dynamics of electrical circuits are considered. It is shown that the results of the theoretical and numerical analyses of these models are consistent.

翻译：获得了两种求解时变半线性微分代数方程的联合数值方法，并证明了方法的收敛性与正确性。在构造方法时，使用了可通过数值计算得到的时变谱投影，这使得能够以原始形式对方程进行数值求解，无需额外的解析变换。为提升第二种方法的精度，引入了重计算技术。所开发的方法适用于非线性部分连续但可能不可微的时变微分代数方程，且在关于方程全局可解性及方法收敛性的定理中，未采用全局Lipschitz条件等限制。这拓展了方法的适用范围。全局可解性定理条件的满足确保了任意给定时间区间上存在唯一精确解，从而也可在任意时间区间上寻找近似解。通过数值算例展示了方法在不同场景下的能力与有效性，并以电路动力学数学模型为例，证实了理论分析与数值结果的一致性。