Global Error Control in Adaptive Nordsieck Methods

SIAM Journal on Scientific Computing, 34(2) (2012), A839-A860
http://dx.doi.org/10.1137/100791932

This paper deals with variable-stepsize Nordsieck formulas applied to ordinary differential equations. It focuses on local and global error evaluation techniques in the mentioned numerical schemes. The error estimators are derived for both consistent Nordsieck methods and quasi-consistent ones. It is also shown how quasi-consistent Nordsieck formulas, which suffer on variable grids from the order reduction phenomenon, can be modified in an optimal way to avoid the order reduction. The latter results in a new class of reducible Nordsieck methods designed here. In addition, we supply our numerical schemes with an automatic global error control mechanism to arrive at the adaptive numerical technique for solving differential equations of preassigned quality. The capacity for deriving numerical solutions satisfying user-supplied accuracy conditions in automatic mode is confirmed by numerical experiments on two test problems.