Abstract It is well known that the seven-step backward difference formula (BDF) is unstable for parabolic equations, since it is not even zero-stable. However, a linear combination of two non zero-stable schemes, namely the seven-step BDF and its shifted counterpart, can yield A () stability. Based on this observation the authors in Akrivis, Chen, and Yu (IMA J. Numer. Anal. , (2025), 3207–3234) proposed the weighted and shifted seven-step BDF methods for parabolic equations, whose stability regions are larger than the stability regions of the standard BDF. Nonetheless, this approach is not directly applicable to parabolic equations with nonsmooth data, and may suffer from severe order reduction. This motivates us to design proper correction time-stepping schemes to restore the desired kth-order convergence rate of the k-step weighted and shifted BDF (k 7) convolution quadrature for parabolic problems. We prove that the desired kth-order convergence can be recovered even if the source term is discontinuous and the initial value is nonsmooth data. Numerical experiments illustrate the theoretical results.
Liu et al. (Fri,) studied this question.