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.

翻译：我们为基于经典BDF2公式的变步长延迟校正（DC）方法建立了新的理论框架。通过利用离散正交卷积核，根据近期稳定性定义（SINUM, 60: 2253-2272），证明了若干高阶BDF2-DC方法在任意时间网格上具有稳定性。这显著放宽了现有BDF2-DC方法（BIT, 62: 1789-1822）的步长比限制。通过考虑起始近似值的数值影响，建立了相应的精确误差估计，结果表明BDF2-DC方法无后效性，即BDF2格式的低阶起始方案不会导致高阶BDF2-DC方法的精度损失。在分级网格和随机时间网格上的大量数值实验验证了该新理论。