Combined methods for solving time-varying semilinear differential-algebraic equations with the use of spectral projectors and applications
This work addresses numerical challenges in solving DAEs for applications like electric circuit modeling, but it appears incremental as it builds on existing spectral projector techniques.
The paper tackles the problem of solving time-varying semilinear differential-algebraic equations (DAEs) by developing two combined numerical methods that use spectral projectors, enabling solution without analytical transformations. It proves convergence and correctness, extends applicability by not requiring global Lipschitz conditions, and demonstrates effectiveness through numerical examples and applications to electric circuit models.
Two combined methods for computing solutions of time-varying semilinear differential-algebraic equations (descriptor systems) are obtained. When constructing the methods, time-varying spectral projectors which can be found numerically are used. This enables one to numerically solve the differential-algebraic equation (DAE) in the original form without additional analytical transformations. The convergence and correctness of the developed methods are proved. The methods are applicable to the semilinear DAEs with the continuous nonlinear part which may not be differentiable in time. The global Lipschitz condition and other conditions of this kind are not used in the presented theorems on the global solvability of DAEs and on the convergence of the methods. This extends the scope of the methods. The obtained theorems ensure both the existence of a unique global exact solution and the convergence of the methods, which enables one to compute an approximate solution on any given time interval. Numerical examples illustrating the capabilities of the methods and their effectiveness in various situations are provided. To demonstrate the practical application of the obtained methods and theorems, the numerical and theoretical analyses of mathematical models of the dynamics of electric circuits are carried out. It is shown that their results are consistent.