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

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.


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
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