Correction of weighted and shifted seven-step BDF for parabolic equations with nonsmooth data

Abstract

It is well known that the seven-step backward difference formula (BDF) is unstable for the 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 $A(\hat{I}\pm )$ stable. Based on this observation, the authors [Akrivis, Chen, and Yu, IMA J. Numer. Anal., DOI:10.1093/imanum/drae089] propose the weighted and shifted seven-step BDF methods for the parabolic equations, which stability regions are larger than the standard BDF. Nonetheless, this approach is not directly applicable for the parabolic equations with nonsmooth data, which may suffer from severe order reduction. This motivates us to design proper correction time-stepping schemes to restore the desired $k$th-order convergence rate of the $k$-step weighted and shifted BDF ($k\le 7$) convolution quadrature for the parabolic problems. We prove that the desired $k$th-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.