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Analysis of adaptive BDF2 scheme for diffusion equations
Hong-lin Liao, Zhimin Zhang
TL;DR
Adaptive BDF2 on unequal time meshes has challenging stability and convergence analysis. The paper develops a convolution-kernel framework and proves stability and convergence results under step-ratio conditions, including discrete energy dissipation for dissipative problems.
Problem
Stability and convergence theory for adaptive multistep methods with unequal time steps remains challenging, including for the linear heat equation.
Source: §1 · p. 1Method
The analysis uses positive semi-definite BDF2 convolution kernels and discrete orthogonal convolution kernels to study adaptive BDF2 for linear reaction-diffusion equations.
Source: §1 · pp. 2–3Results
Under S1, adaptive BDF2 is unconditionally stable in L2 and possibly first-order convergent; for dissipative problems it preserves the discrete energy dissipation law.
Source: §1 · pp. 3–4Takeaways & Limitations
S1 permits increasing time steps by factors up to 3.561, while second-order convergence is established when only a small set of ratios falls outside the stricter regime.
Source: §1 · p. 4The energy estimate gives a suboptimal H1-seminorm error estimate in temporal accuracy and dependence on the diffusion coefficient, especially when ε is small.
Source: §2.1 · p. 8
Abstract
from arXiv · showhide
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.
1 Introduction
Adaptive multistep methods are challenging to analyze on unequal time meshes, and BDF2’s stability and convergence theory remains incomplete. The introduction frames a kernel-based analysis for linear reaction-diffusion equations.
1 Introduction
- The analysis treats BDF2 as a discrete convolution and combines positive semi-definiteness of its kernels with discrete orthogonal convolution kernels.The orthogonal kernels are introduced to establish L2 stability and convergence.Source: §1 · pp. 2–3
- Under S1, adaptive BDF2 preserves the discrete energy dissipation law and yields an L2 estimate that supports monotonicity, unconditional stability, and possibly first-order convergence.The estimate is described as simulating the continuous estimate.Source: §1 · pp. 3–4
- For bounded reaction coefficients, S1 also gives unconditional L2 stability and ensures zero-stability.The introduction presents this as an update to the classical stability restriction.Source: §1 · p. 4
- S1 allows increasing time steps with amplification factors up to 3.561 and restricts amplification in fast-to-slow transitions.The introduction notes that very large time steps can reduce numerical precision.Source: §1 · p. 4
- Second-order L2 convergence is established under S2, which keeps ratios in S1 while requiring the exceptional-index set Rp to be small relative to N.The condition is stated as |Rp| = N0 ≪ N.Source: §1 · p. 4
- For second-order computations, users are advised to choose all step ratios in (0, 1 + √2), while S2 allows some redundancy in practical adaptive choices.The introduction characterizes S2 as more theoretical than practical and mentions random-mesh tests supporting the analysis.Source: §1 · p. 4
2 Stability analysis for dissipative diffusion problem
For dissipative problems with κ ≤ 0, the analysis establishes discrete energy dissipation and unconditional stability in both the energy norm and the L2 norm under the stated kernel or step-ratio conditions.
2.1 Positive semi-definiteness and energy stability
- S1 is sufficient but not necessary: the authors note that kernel positive semi-definiteness can hold beyond it, while a weaker combined-ratio condition remains open.The proof restricts individual step ratios rather than establishing a condition combining all ratios.Source: §2.1 · pp. 5–6
- Under S1 and κ ≤ 0, the scheme satisfies a discrete energy-dissipation law and is unconditionally stable in the energy norm.With zero exterior force, it preserves the energy dissipation law at discrete time levels.Source: §2.1 · pp. 6, 8
- When ∥∂tu∥ is large, small time steps are needed to capture solution behavior; when it is small, larger steps can accelerate integration.The discrete energy form motivates this step-size choice.Source: §2.1 · p. 8
2.2 Orthogonal convolution kernels and L2 norm stability
- Positive semi-definiteness of the BDF2 kernels, or S1, is sufficient for unconditional L2 stability; with f ≡ 0, the scheme also preserves L2 norm monotonicity.The L2 estimate requires neither a discrete Grönwall inequality nor a maximum-step-size restriction.Source: §2.2 · pp. 10–11
- The L2 analysis applies DOC kernels and their orthogonal identity, avoiding discrete Grönwall inequalities and a maximum-step-size restriction.The resulting estimate recovers the continuous solution estimate under S1.Source: §2.2 · pp. 9, 11
- Second-order accuracy follows under S1 when both the first-level solution and its discrete time derivative are second-order accurate.The energy estimate itself yields a suboptimal H1-seminorm error estimate in temporal accuracy and in its dependence on ε, especially for small ε.Source: §2.1 · p. 8, §2.2 · p. 12
3 L2 norm convergence for linear diffusion problems
The section establishes L2 stability and convergence for adaptive BDF2 under step-ratio and kernel conditions, while showing that convergence order depends on the starting scheme and time mesh.
3.1 Priori estimate
- When the BDF2 kernels are positive semi-definite (or sufficient condition S1 holds) and the maximum time-step size τ ≤ 1/(4κ*), the discrete solution u_n of scheme (1.3) is unconditionally stable in the L2 norm.Source: §3.1 · p. 13
- Condition S1 also ensures zero-stability of BDF2 for the nonlinear ODE model with Lipschitz-continuous perturbations.The paper describes S1 as updating the Grigorieff stability restriction.Source: §3.1 · p. 14
3.2 Consistency and convergence
- With positive semi-definite BDF2 kernels or S1, and τ ≤ 1/(4κ*, the solution converges in L2, with the analysis identifying first-order convergence.The theorem requires the stated maximum-step-size bound; the text notes that the convolution term can be unbounded as step sizes vanish while the solution remains first-order convergent.Source: §3.2 · p. 17
- Second-order accuracy can be recovered by using a high-order starting scheme or choosing τ1 = O(τ^2).The paper attributes the accuracy loss to unequal time steps and their associated step ratios; its BDF1 starting scheme is only first-order consistent.Source: §3.2 · pp. 14, 15, 17
4 Numerical example
The example tests nonuniform BDF2 on a heat equation with an exact solution, using randomly generated time steps. The reported tests show robust stability and second-order convergence when large step ratios are infrequent.
4 Numerical example
- The tests solve a forced heat equation on (0, 2)^2 × (0, 1] with exact solution u = e^-t sin 2πx cos 2πy, using random time steps.The spatial discretization is Fourier pseudo-spectral; the forcing is chosen to admit the stated exact solution.Source: §4 · p. 18
- When large step ratios occurred in about 5% of time levels, the adaptive BDF2 method was robustly stable and second-order convergent in the reported tests.The authors report this pattern from the displayed data and additional tests not listed.Source: §4 · p. 18, Table 4.1 · p. 18, Table 4.2 · p. 18
5 Concluding remarks
The concluding section presents discrete orthogonal convolution kernels as a framework for analyzing adaptive BDF2. It connects the approach to stability and convergence analysis and suggests applications to other convolution-form time approximations.
5 Concluding remarks
- The framework introduces a new class of discrete orthogonal convolution kernels for numerical analysis.The paper also presents an alternative form obtained by exploiting their orthogonality.Source: §5 · p. 19
- For adaptive BDF2 on linear reaction-diffusion equations, the approach gives a concise stability and convergence theory described as similar to that of BDF1.The authors expect the framework to help establish optimal L2 error estimates for other nonlocal time approximations with a discrete convolution form.Source: §5 · p. 19