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Two combined numerical methods for solving time-varying semilineardifferential-algebraic equations (DAEs) are obtained. The convergence andcorrectness of the methods are proved. When constructing the methods,time-varying spectral projectors which can be found numerically are used. Thisenables to numerically solve the DAE in the original form without additionalanalytical transformations. To improve the accuracy of the second method,recalculation is used. The developed methods are applicable to the DAEs withthe continuous nonlinear part which may not be differentiable in time, and therestrictions of the type of the global Lipschitz condition are not used in thepresented theorems on the DAE global solvability and the convergence of themethods. This extends the scope of methods. The fulfillment of the conditionsof the global solvability theorem ensures the existence of a unique exactsolution on any given time interval, which enables to seek an approximatesolution also on any time interval. Numerical examples illustrating thecapabilities of the methods and their effectiveness in various situations areprovided. To demonstrate this, mathematical models of the dynamics ofelectrical circuits are considered. It is shown that the results of thetheoretical and numerical analyses of these models are consistent.

Combined numerical methods for solving time-varying semilinear  differential-algebraic equations with the use of spectral projectors and  recalculation