Communications in Mathematical Sciences

Volume 21 (2023)

Number 3

Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations

Pages: 775 – 794



Chengchao Zhao (Beijing Computational Science Research Center, Beijing, China)

Nan Liu (School of Mathematics and Statistics, Wuhan University, Wuhan, China)

Yuheng Ma (School of Mathematics and Statistics, Wuhan University, Wuhan, China)

Jiwei Zhang (School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science,Wuhan University, Wuhan, China)


In this paper we consider a linearized variable-time-step two-step backward differentiation formula (BDF2) scheme for solving nonlinear parabolic equations. The scheme is constructed by using the variable time-step BDF2 for the linear term and a Newton linearized method for the nonlinear term in time combining with a Galerkin finite element method (FEM) in space. We prove the unconditionally optimal error estimate of the proposed scheme under mild restrictions on the ratio of adjacent time-steps, i.e. the ratio less than $4.8645$, and on the maximum time step. The proof involves the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and the error splitting approach. In addition, our analysis also shows that the first level solution obtained by BDF1 (i.e., backward Euler scheme) does not cause the loss of global accuracy of second order. Numerical examples are provided to demonstrate our theoretical results.


nonlinear parabolic equations, variable time-step BDF2, orthogonal convolution kernels, stability and convergence, error splitting approach

2010 Mathematics Subject Classification

65M06, 65M12

Jiwei Zhang is supported in the NSFC under grants nos. 12171376 and 2020-JCJQ-ZD-029, and by the Fundamental Research Funds for Central Universities, grant no. 2042021kf0050.

Received 22 January 2022

Received revised 23 July 2022

Accepted 5 August 2022

Published 27 February 2023