Variable Steps Weighted And Shifted BDF3 Methods

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,BIT.42(2002)689-701]under the condition rk:=?k/?k-1<1.199,where rk 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 rk?1.405 are proved.Meanwhile,we show that the upper bound of the ratio rk is less than(?)for BDF3 scheme.In addition,based on the idea of[Wang and Ruuth,J.Comput.Math.26(2008)838-855;Chen,Yu,and Zhang,SIAM J.Numer.Anal.,Major Revised,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 rk?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.