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

Two combined numerical methods for solving time-varying semilinear differential-algebraic equations (DAEs) are obtained. These equations are also called degenerate DEs, descriptor systems, operator-differential equations and DEs on manifolds. The convergence and correctness of the methods are proved. When constructing methods we use, in particular, time-varying spectral projectors which can be numerically found. This enables to numerically solve and analyze the considered DAE in the original form without additional analytical transformations. To improve the accuracy of the second method, recalculation (a ``predictor-corrector'' scheme) is used. Note that the developed methods are applicable to the DAEs with the continuous nonlinear part which may not be continuously differentiable in $t$, and that the restrictions of the type of the global Lipschitz condition, including the global condition of contractivity, are not used in the theorems on the global solvability of the DAEs and on the convergence of the numerical methods. This enables to use the developed methods for the numerical solution of more general classes of mathematical models. For example, the functions of currents and voltages in electric circuits may not be differentiable or may be approximated by nondifferentiable functions. Presented conditions for the global solvability of the DAEs ensure the existence of an unique exact global solution for the corresponding initial value problem, which enables to compute approximate solutions on any given time interval (provided that the conditions of theorems or remarks on the convergence of the methods are fulfilled). In the paper, the numerical analysis of the mathematical model for a certain electrical circuit, which demonstrates the application of the presented theorems and numerical methods, is carried out.