A stochastic approach to global error estimation in ODE multistep numerical integration (Q917235)

The numerical integration of ordinary differential equations (ODE) related to initial-value problems, especially in the situations of very long intervals of integration is studied. A stochastic approach to estimate the global errors is proposed. The authors consider local truncation and round-off errors. In the procedure proposed, global errors are treated with respect to their order of magnitude. They are represented, or modelled, through the distribution of random variables belonging to stochastic sequences, which take into account the influence both of local truncation and round-off errors. The dispersions of these random variables, in terms of their variances, are assumed to give an estimation of the errors. The error estimation procedure is developed for Adams-Bashforth-Moulton (ABM) type of multistep methods. The numerical integrator is formulated based on the ABM multistep method. The computational effort in integrating the variational equations to propagate the error covariance matrix associated with error magnitudes and correlations is minimized by employing a first or second order Euler method. The diagonal variances of the covariance matrix, derived using the stochastic approach developed in this paper, are found to furnish reasonable precise measures of the order of magnitude of accumulated global errors in short-term as well as long-term orbit propagations.