Weighted and shifted BDF3 methods on variable grids for a parabolic problem

As is well known, the stability of the 3-step backward differentiation formula (BDF3) on variable grids for a parabolic problem is analyzed in [Calvo and Grigorieff, \newblock BIT. \textbf{42} (2002) 689--701] under the condition $r_k:=\tau_k/\tau_{k-1}<1.199$, where $r_k$ is the adjacent time-step ratio. In this work, we establish the spectral norm inequality, which can be used to give a upper bound for the inverse matrix. Then the BDF3 scheme is unconditionally stable under a new condition $r_k\leq 1.405$. Meanwhile, we show that the upper bound of the ratio $r_k$ is less than $\sqrt{3}$ for BDF3 scheme. In addition, based on the idea of [Wang and Ruuth, J. Comput. Math. \textbf{26} (2008) 838--855; Chen, Yu, and Zhang, arXiv:2108.02910], we design a weighted and shifted BDF3 (WSBDF3) scheme for solving the parabolic problem. We prove that the WSBDF3 scheme is unconditionally stable under the condition $r_k\leq 1.771$, which is a significant improvement for the maximum time-step ratio. The error estimates are obtained by the stability inequality. Finally, numerical experiments are given to illustrate the theoretical results.