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Mathematics > Numerical Analysis

arXiv:2201.06008 (math)
[Submitted on 16 Jan 2022]

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

Authors:Chengchao Zhao, Nan Liu, Yuheng Ma, Jiwei Zhang
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Abstract: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. $0<r_k < r_{\max} \approx 4.8645$ 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 $u^1$ 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.
Comments: 18 pages, 2 figures
Subjects: Numerical Analysis (math.NA)
MSC classes: 65M06, 65M12
Cite as: arXiv:2201.06008 [math.NA]
  (or arXiv:2201.06008v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2201.06008

Submission history

From: Chengchao Zhao [view email]
[v1] Sun, 16 Jan 2022 09:53:19 UTC (60 KB)
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