Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

TL;DR

This paper introduces two combined numerical methods utilizing spectral projectors and Taylor expansions for solving semilinear differential-algebraic equations, with proven convergence and broad applicability for global solution computation.

Contribution

The paper develops two novel numerical methods for DAEs that require weaker restrictions and enable global solution computation without additional analytical transformations.

Findings

Methods are proven to converge.

Comparative analysis shows effectiveness in various situations.

Methods require weaker restrictions than existing approaches.

Abstract

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.