  Two combined numerical methods for solving semilinear differential-algebraic
equations (DAEs) are obtained and their convergence is proved. The comparative
analysis of these methods is carried out and conclusions about the
effectiveness of their application in various situations are made. In
comparison with other known methods, the obtained methods require weaker
restrictions for the nonlinear part of the DAE. Also, the obtained methods
enable to compute approximate solutions of the DAEs on any given time interval
and, therefore, enable to carry out the numerical analysis of global dynamics
of mathematical models described by the DAEs. The examples demonstrating the
capabilities of the developed methods are provided. To construct the methods we
use the spectral projectors, Taylor expansions and finite differences. Since
the used spectral projectors can be easily computed, to apply the methods it is
not necessary to carry out additional analytical transformations.
