Analysis of adaptive BDF2 scheme for diffusion equations

Liao, Hong-lin; Zhang, Zhimin

doi:10.1090/mcom/3585

Analysis of adaptive BDF2 scheme for diffusion equations
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by Hong-lin Liao and Zhimin Zhang HTML | PDF

Math. Comp. 90 (2021), 1207-1226 Request permission

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 $r_k\coloneq \tau _k/\tau _{k-1}\le (3+\sqrt {17})/2\approx 3.561$, 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.

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