A local-global stepsize control for multistep methods applied to semi-explicit index 1 differential-algebraic equations

Journal of Applied Mathematics and Computing, 6(3) (1999), 463-492


In this paper we develop a new procedure to control stepsize for linear multistep methods applied to semi-explicit index 1 differential-algebraic equations. In contrast to the standard approach, the error control mechanism presented here is based on monitoring and controlling both the local and global errors of multistep formulas. As a result, such methods with the local-global stepsize control solve differential-algebraic equations with any prescribed accuracy (up to round-off errors). For implicit multistep methods we give the minimum number of both full and modified Newton iterations allowing the iterative approximations to be correctly used in the procedure of the local-global stepsize control. We also discuss validity of simple iterations for high accuracy solving differential-algebraic equations. Numerical tests support the theoretical results of the paper.