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Liao, Hong-Lin; Zhang, Zhimin

Analysis of adaptive BDF2 scheme for diffusion equations. (English) Zbl 1466.65067

Math. Comput. 90, No. 329, 1207-1226 (2021).
Summary: 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 \(r_k:=\tau_k/\tau_{k-1}\le (3+\sqrt{17})/2\approx 3.561\), 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.

Cited in 2 Reviews
Cited in 16 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs

Keywords:

linear diffusion equations; adaptive BDF2 scheme; orthogonal convolution kernels; positive semi-definiteness; stability and convergence

Software:

ode45; MATLAB ODE suite; Matlab; ode23s; ode113; Ode15s; ode23
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\textit{H.-L. Liao} and \textit{Z. Zhang}, Math. Comput. 90, No. 329, 1207--1226 (2021; Zbl 1466.65067)
Full Text: DOI arXiv
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