  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.
