Analysis of adaptive BDF2 scheme for diffusion equations
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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:=τ_k/τ_k-1<(3+√(17))/2≈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< 1+√(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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