Analysis of adaptive BDF2 scheme for diffusion equations

TL;DR

This paper provides a new theoretical analysis of the adaptive BDF2 scheme for diffusion equations, establishing stability and convergence conditions based on time-step ratios, and demonstrating energy and norm preservation for dissipative problems.

Contribution

It introduces a novel framework for analyzing adaptive BDF2 schemes, deriving stability and convergence criteria related to time-step ratios, and confirms energy dissipation and monotonicity properties.

Findings

Stable for time-step ratios up to ~3.561

Second-order convergence with small ratios or high-order schemes

Energy dissipation and norm monotonicity preserved in dissipative cases

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 $r_k:=\tau_k/\tau_{k-1}\le(3+\sqrt{17})/2\approx3.561$, 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.