Computational Mathematics and Mathematical Physics, 44(8) (2004), 1314-1333

To numerically solve systems of ordinary differential equations, a theory is developed for multistep methods with polynomial interpolation of the numerical solution on nonuniform grids. Stability and convergence are examined, and the evaluation and control of asymptotically correct estimates for local and global errors are discussed. It is shown that interpolation multistep methods with local–global
stepsize control are capable of ensuring any prescribed accuracy in the automatic mode (provided that no round-off errors are made). This class of methods is theoretically substantiated and verified on test problems.