Analysis of adaptive BDF2 scheme for diffusion equations

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 $r_k:={\tau }_k/{\tau }_{k-1}\le (3+\sqrt{17})/2\approx 3.561$, 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.

20 pages

References in corpus (2)

• On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation
• An adaptive BDF2 implicit time-stepping method for the phase field crystal model

Cited by in corpus (14)

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• An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation
• Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn-Hilliard model
• Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection
• Stability of variable-step BDF2 and BDF3 methods
• The variable-step L1 scheme preserving a compatible energy law for time-fractional Allen-Cahn equation
• Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators
• Compatible $L^2$ norm convergence of variable-step L1 scheme for the time-fractional MBE mobel with slope selection
• $L^2$ norm error estimates of BDF methods up to fifth-order for the phase field crystal model
• Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model
• Average energy dissipation rates of explicit exponential Runge-Kutta methods for gradient flow problems
• An efficient two-grid fourth-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation
• Energy dissipation law and maximum bound principle-preserving linear BDF2 schemes with variable steps for the Allen-Cahn equation
• Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods