Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 57, Issue 1, Pages 495-525Publisher
SIAM PUBLICATIONS
DOI: 10.1137/18M1206084
Keywords
variable step BDF2 scheme; convergence analysis; Cahn-Hilliard equation
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Funding
- National Natural Science Foundation of China [11671098, 11331004, 91630309]
- Ministry of Education of China
- State Administration of Foreign Experts Affairs of China under a 111 project [B08018]
- National Natural Science Foundation of China
- Southern University of Science and Technology
- Shanghai Center for Mathematical Sciences at Fudan University
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We present and analyze a second order in time variable step BDF2 numerical scheme for the Cahn-Hilliard equation. The construction relies on a second order backward difference, convex-splitting technique and viscous regularizing at the discrete level. We show that the scheme is unconditionally stable and uniquely solvable. In addition, under mild restriction on the ratio of adjacent time-steps, an optimal second order in time convergence rate is established. The proof involves a novel generalized discrete Gronwall-type inequality. As far as we know, this is the first rigorous proof of second order convergence for a variable step BDF2 scheme, even in the linear case, without severe restriction on the ratio of adjacent time-steps. Results of our numerical experiments corroborate our theoretical analysis.
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