Extrapolation multistep methods for numerical solution of second-order differential algebraic equations

In the article, we consider linear second-order differential algebraic equations (DAEs) on a finite interval of integration with given initial data. A class of problems with a unique sufficiently smooth solution is identified in terms of matrix polynomials. We assume that the solution to the problem may contain rigid and rapidly oscillating components. This paper highlights the main challenges of developing algorithms for numerical solutions to the class of problems under consideration. We propose to represent the original problem in the form of a system of integral differential or integral equations with an identically degenerate matrix in front of the main part for constructing effective methods of numerical solution of second-order DAEs. Moreover, we construct numerical solution methods for problems represented in this way. These algorithms are based on explicit Adams quadrature formulas for calculating the integral term and on extrapolation formulas for other terms. The results of test example calculations are presented and analyzed.