Estimating Global Errors in Time Stepping*

This study introduces new strategies for global error estimation in time-stepping algorithms. The new methods propagate the defect along with the numerical solution much like the Zadunaisky procedure; however, the proposed approach allows for overlapped internal computations and, therefore, represents a generalization of the classical numerical schemes for solving differential equations with global error estimation. The resulting algorithms can be effectively represented as general linear methods. We present a few explicit self-starting schemes akin to Runge-Kutta methods with global error estimation and illustrate the theoretical considerations on several examples.