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. 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.