ANALYSIS OF ADAPTIVE BDF2 SCHEME FOR DIFFUSION EQUATIONS

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) <= (3 + root 17)/2 approximate to 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+ root 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.

Automated summary

In this paper, the variable two-step backward differentiation formula (BDF2) is reexamined using a new theoretical framework, highlighting its stability and convergence properties for linear reaction-diffusion equations. The study shows that under specific conditions, the adaptive BDF2 time-stepping scheme is unconditionally stable and demonstrates (potentially first-order) convergence in the L-2 norm. Additionally, for linear dissipative diffusion problems, the stable BDF2 method is shown to preserve energy dissipation law in the H-1 seminorm and maintain monotonicity in the L-2 norm at discrete levels.