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Analysis of adaptive BDF2 scheme for diffusion equations

Authors :
Zhimin Zhang
Hong-lin Liao
Publication Year :
2019
Publisher :
arXiv, 2019.

Abstract

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k:=\tau_k/\tau_{k-1}\le(3+\sqrt{17})/2\approx3.561$, the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the $L^2$ norm. The second-order temporal convergence can be recovered if almost all of time-step ratios $r_k\le 1+\sqrt{2}$ or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the $H^1$ seminorm) and the $L^2$ norm monotonicity at the discrete levels. An example is included to support our analysis.<br />Comment: 20 pages

Details

Database :
OpenAIRE
Accession number :
edsair.doi.dedup.....1b5f9884b6e081fb382c68d90b5d8636
Full Text :
https://doi.org/10.48550/arxiv.1912.11182