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.

中文翻译：

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扩散方程的自适应BDF2方案分析

利用 BDF2 卷积核的半正定性和一类正交卷积核，通过新的理论框架重新审视可变两步向后微分公式 (BDF2)。我们证明，如果相邻时间步长比 $r_k:=\tau_k/\tau_{k-1}\le(3+\sqrt{17})/2\approx3.561$，则自适应 BDF2 时间步长线性反应扩散方程的方案是无条件稳定的，并且（可能是一阶）收敛于 $L^2$ 范数。如果使用几乎所有的时间步长比 $r_k\le 1+\sqrt{2}$ 或一些高阶起始方案，则可以恢复二阶时间收敛。特别地，对于线性耗散扩散问题，稳定的 BDF2 方法在离散水平上保留了能量耗散定律（在 $H^1$ 半范数中）和 $L^2$ 范数单调性。