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

TL;DR

This paper introduces two new numerical methods for solving time-varying semilinear differential-algebraic equations using spectral projectors, allowing solutions directly in the original form without extra transformations, and proves their convergence and correctness.

Contribution

The paper develops combined methods utilizing spectral projectors for DAEs, extending applicability to non-differentiable nonlinear parts and removing the need for Lipschitz conditions, with proven convergence and global solvability.

Findings

Methods successfully solve DAEs in original form.

Proven convergence and uniqueness of solutions.

Numerical examples confirm effectiveness and applicability.

Abstract

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.