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On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation
Hong-lin Liao, Tao Tang, Tao Zhou
TL;DR
Analysis of nonuniform-grid time stepping remains incomplete, despite its relevance to phase-field models with multiple time scales. This paper analyzes nonuniform BDF2 for Allen–Cahn using kernel recombination and complementary techniques. It establishes energy stability, a discrete maximum principle under a ratio restriction and practical step constraint, and second-order maximum-norm convergence.
Problem
Numerical analysis for nonuniform time grids remains incomplete, although phase-field solutions can evolve on fast and slow time scales.
Source: §1 · pp. 1–2Method
The paper analyzes nonuniform BDF2 for the Allen–Cahn equation and develops a kernel recombination and complement technique.
Source: §1 · p. 2Results
The scheme is energy stable under a ratio restriction and preserves the discrete maximum principle under a ratio restriction and practical time-step constraint, with second-order maximum-norm convergence.
Source: §1 · p. 2, §2.2 · p. 4Takeaways & Limitations
The discrete maximum principle supports maximum-norm convergence analysis without a Lipschitz assumption on the nonlinear bulk force.
Source: §2.2 · p. 4Whether the ratio constraints S1 and S0 are optimal remains an open question.
Source: §7 · p. 22
Abstract
from arXiv · showhide
In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=\tau_k/\tau_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.
1 Introduction
Phase-field models have multiple time scales, motivating nonuniform time grids, but numerical analysis for nonuniform grids remains incomplete. This work studies whether nonuniform BDF2 preserves the Allen–Cahn equation’s energy dissipation and maximum-principle properties discretely.
1 Introduction
- Nonuniform grids address phase-field solutions’ fast initial dynamics and slower later coarsening, while their numerical analysis remains less developed than for uniform grids.Source: §1 · pp. 1–2
- The scheme is energy stable under a time-step-ratio restriction.Source: §1 · p. 2
- The nonuniform BDF2 scheme preserves the discrete maximum principle under a time-step-ratio restriction and a practical time-step constraint.The authors describe this as the first such result for the scheme.Source: §1 · p. 2
- The analysis establishes second-order convergence in the maximum norm, supported by numerical experiments.Source: §1 · p. 2
- A kernel recombination and complement technique replaces the generalized discrete Gronwall approach used in related Cahn–Hilliard work.The related work combines nonuniform BDF2 with convex splitting and presents energy stability and convergence analysis under similar ratio restrictions.Source: §1 · p. 2
2 Preliminaries
The paper sets up the Allen–Cahn equation on a periodic spatial domain and discretizes time with nonuniform BDF2. It analyzes energy stability, maximum-principle preservation, and maximum-norm convergence, using kernel recombination and complementary techniques.
2 Preliminaries
- The Allen–Cahn setup uses f(u)=u^3−u, interaction length ε, and periodic boundary conditions on a square domain.Source: §2 · p. 2
2.1 The nonuniform BDF2 scheme
- The time grid permits varying step sizes, with adjacent-step ratios defined for the nonuniform BDF2 formulation.Source: §2.1 · p. 3
- The nonuniform BDF2 scheme uses a BDF1 step to generate its first-level solution before applying the two-step formula.Source: §2.1 · p. 3
2.2 Summary of main contributions
- The nonuniform BDF2 scheme is energy stable under the time-step ratio condition S1.Source: §2.2 · p. 4
- KRC recombines the original variables into new variables and reformulates BDF2 as a convolution with nonnegative, monotonically decreasing kernels.Source: §2.2 · p. 4
- The scheme preserves the discrete maximum principle under the Grigorieff zero-stability ratio condition, and yields second-order maximum-norm convergence under S0 without a Lipschitz assumption on the nonlinear bulk force.The convergence analysis uses the complementary stage of KRC and a new discrete Gronwall inequality.Source: §2.2 · p. 4
3 Solvability and energy stability
The section establishes unique solvability and energy stability for the nonuniform BDF2 scheme. Its energy argument uses a modified discrete energy and properties of the discrete Laplacian.
3 Solvability and energy stability
- The discrete Laplacian matrix is symmetric, negative semidefinite, and diagonally dominant, properties used in the stability analysis.The proof takes an L2 inner product of the scheme with the time difference.Source: §3 · pp. 4–6
- The scheme is uniquely solvable under its stated time-step condition, with strict convexity of the objective function implying uniqueness.The condition is also described as practical, since requiring τ_n < 1 suffices.Source: §3 · p. 5
- The modified discrete energy equals the original discrete energy initially and approaches it as τ tends to zero.The modified energy is introduced for the stability analysis.Source: §3 · p. 5
- Under S1 and the theorem’s additional time-step condition, the discrete solution satisfies the stated energy-stability estimate.The supplied theorem passages do not include the estimate’s full expression.Source: §3 · p. 5
4 Kernels recombination and discrete maximum principle
The section reformulates BDF2 using recombined variables and kernels, then uses a parameter choice and time-step condition to establish a discrete maximum principle. The maximum-principle result is presented under the step-ratio restriction S0.
4 Kernels recombination and discrete maximum principle
- The recombination and complementary technique is used to analyze both the discrete maximum principle and convergence estimates for scheme (2.7).The reformulated BDF2 formula is written as a discrete convolution using new variables and kernels.Source: §4.1 · pp. 7–8
4.1 Reformation of BDF2 formula
- The variables ¯v_k = v_k − ηv_{k−1} recast BDF2 in convolution form, with the resulting kernels required to be nonnegative and decreasing.S0 is identified with Grigorieff’s zero-stability condition.Source: §4.1 · pp. 7–8
4.2 Choice of recombined parameter
- Lemma 4.1 imposes a time-step-size condition and a constraint on η for a stated property of the matrix Q(n).The passage introducing the lemma names these conditions but does not include the full matrix property.Source: §4.2 · p. 9
4.3 Discrete maximum principle
- Under S0 and the stated time-step-size condition, BDF2 preserves the discrete maximum principle; η is analytical and is not needed in computation.For interface tracking, the text notes that h = O(ε), and that the time-step restriction is approximately equivalent to a simpler condition.Source: §4.3 · pp. 11–12
5 Complementary kernels and convergence analysis
The section develops a discrete Gronwall lemma using complementary convolution kernels and applies it in a maximum-norm convergence analysis. The stated theorem assumes smooth, bounded initial data, S0, and a time-step constraint.
5 Complementary kernels and convergence analysis
- The complementary kernels and their properties are used to build a discrete Gronwall lemma for analysis of the nonuniform BDF2 scheme.The paper notes that this type of discrete kernel was previously introduced for Caputo derivatives and later generalized.Source: §5 · pp. 12–13
- Under the theorem’s smoothness, initial-data, step-ratio, and time-step assumptions, the numerical solution converges in the maximum norm.The stated error-bound constant is independent of time-step sizes and time-step ratios.Source: §5 · p. 14
- The error equation separates time and space truncation errors after subtracting the numerical scheme from the equation for the exact solution.The equation includes Υ^n and R^n as the time- and space-truncation errors, respectively.Source: §5 · p. 14
6 Numerical implementations
The numerical examples test adaptive and uniform time stepping, accuracy on random meshes, and Allen–Cahn bubble dynamics. They report efficient adaptive stepping, second-order accuracy on random meshes, and bounded maximum norm and energy behavior under specified settings.
6 Numerical implementations
- Each nonlinear BDF2 time level is solved by Newton-type iteration, initialized from the previous solution and stopped at tolerance 10^-12.The implementation uses the previous-level solution as the initial iterate.Source: §6 · p. 16
6.1 Adaptive time-stepping strategy
- The adaptive strategy compares backward Euler and adaptive BDF2 solutions, using the relative error to guide time-step selection.The stated defaults are safety coefficient ρ = 0.6, tolerance 10^-4, τmax = 0.1, and τmin = 10^-3.Source: §6.1 · p. 17
6.2 Numerical examples
- On random meshes, nonuniform BDF2 retains second-order accuracy even when large step ratios do not satisfy the stated requirement.The experiment uses the maximum-norm error and reports step-ratio counts in Table 1.Source: §6.2 · p. 18
- Adaptive bubble-merging snapshots show four initially separated bubbles coalescing into one while the volume decreases.The simulations report that the Allen–Cahn model does not conserve initial volume.Source: §6.2 · p. 18, Figure 2 · p. 19
- 511 adaptive steps produced an energy curve practically indistinguishable from the small-step uniform run, which required 30,000 steps.Both strategies simulate bubble merging to T = 30.Source: §6.2 · p. 18, Figure 1 · p. 19
- For uniform time steps τ = 0.2 and 0.4, the maximum norm stays bounded by 1 and the discrete energy dissipates.The authors describe the time-step constraint as sufficient for the discrete maximum principle.Source: §6.2 · p. 19, Figure 4 · p. 20
- During coarsening, the discrete maximum stays bounded by 1, energy decays monotonically, and the adaptive strategy is reported as effective.The grain count decreases over time as phase boundaries migrate, decompose, and merge.Source: §6.2 · pp. 19–20
7 Concluding remarks
The conclusion establishes energy stability, a discrete maximum principle, and second-order maximum-norm convergence for variable-step BDF2. It also identifies time-fractional phase-field extensions and the optimality of the ratio constraints as open directions.
7 Concluding remarks
- The adaptive-time coarsening illustrations show solution snapshots at t = 1, 10, 20, 50, 80, and 100, alongside maximum-norm, energy, and time-step histories.Source: Figure 5 · p. 21, Figure 6 · p. 21
- Variable time steps accommodate Allen-Cahn solutions that pass through different regimes requiring finer or coarser steps.The step-size ratio may vary smoothly, though practical computations may require upper bounds.Source: §7 · p. 21
- The discrete maximum principle holds under a time-ratio restriction that coincides with Grigorie’s zero-stability condition.The maximum principle also permits error estimates without Lipschitz assumptions on the nonlinear bulk force.Source: §7 · p. 21
- The nonuniform BDF2 scheme preserves the energy dissipation law under a mild time-ratio constraint.The authors establish this with an appropriate energy method.Source: §7 · p. 21
- A second-order convergence rate in the maximum norm is established using the KRC technique and a new Gronwall inequality.Source: §7 · p. 21
- The KRC technique is expected to extend to more general nonlinear problems.Source: §7 · p. 21
- Developing nonuniform BDF2 schemes for time-fractional phase-field equations remains a challenge, and whether ratio constraints S1 and S0 are optimal is unresolved.Variable steps are especially relevant for time-fractional operators because they require solution information at all time levels.Source: §7 · p. 22