  We provide a new theoretical framework for the variable-step deferred
correction (DC) methods based on the well-known BDF2 formula. By using the
discrete orthogonal convolution kernels, some high-order BDF2-DC methods are
proven to be stable on arbitrary time grids according to the recent definition
of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing
step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The
associated sharp error estimates are established by taking the numerical
effects of the starting approximations into account, and they suggest that the
BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme
for the BDF2 scheme will not cause a loss in the accuracy of the high-order
BDF2-DC methods. Extensive tests on the graded and random time meshes are
presented to support the new theory.
