Energy-rate equidistribution for adaptive SSAV--BDF2 integration of gradient flows

1 Introduction

• Variable-step BDF2 loses the telescoping energy identity, and SAV/SSAV coupling adds consistency errors.Source: §1 · p. 2
• Energy-rate step selection equidistributes a monitor along the energy curve, and g = |Ė| asymptotically minimizes the leading maximum symmetric-energy discrepancy for a fixed step count.The analysis also interprets β as the ratio of the largest permitted step to the energy-error tolerance.Source: §1 · p. 2
• The combined monitor uses g = |dot E| during rapid dissipation and near equilibrium, capping step growth where the first rule becomes too loose and bounding symmetric-energy discrepancy by user-defined τ1/2 in both regimes.Source: §1 · p. 2
• Accepted SSAV–vBDF2 steps satisfy symmetric-energy monotonicity, yielding a physical-energy bound uniform in step count and second-order accuracy under standard step-ratio restrictions.Source: §1 · p. 3
• The method is tested on Cahn–Hilliard, Ohta–Kawasaki, Phase–Field Crystal, and functionalized Cahn–Hilliard problems against fixed-step SSAV–BDF2 at matched accuracy.Source: §1 · p. 3

2.1 Continuous Problem

• The continuous problem uses an energy split into linear and nonlinear parts, with differentiability and positivity assumptions on the nonlinear energy; gradient-dependent nonlinear terms can be handled through its full variational derivative.For pointwise nonlinearities, F is assumed to have bounded second and third derivatives on the relevant solution range and to be bounded below after a shift.Source: §2.1 · p. 3

2.2 SAV Reformulation

• The SAV reformulation introduces an auxiliary variable and yields a modified energy that is nonincreasing, coinciding with the physical energy when w² = Enl(u) + C.For gradient-dependent nonlinear energies, the construction uses the full variational derivative and a problem-specific splitting.Source: §2.2 · pp. 3–4, §2.1 · p. 3

2.3 Variable-Step BDF2 Discretization

• The discretization combines stabilized SAV with variable-step BDF2 in a linearly implicit scheme, and the Laplacian dissipative direction also preserves mass under periodic or homogeneous Neumann boundary conditions.The cited prior work reports that stabilization permits larger steps and that SSAV maintains second-order accuracy in an FCH accuracy test where SAV loses accuracy above Δt > 5 × 10^-4.Source: §2.3 · p. 4

3 Energy-Rate Equidistribution

• Energy-rate monitoring redistributes time points so rapid changes occupy more of the computational interval and are resolved more uniformly.This is a one-dimensional r-adaptivity view: large monitor values produce smaller physical steps, while small values produce larger ones.Source: §3 · p. 5

3.1 Symmetric energy and its relation to dissipation rate

• The leading symmetric-energy discrepancy is proportional to the step size times the absolute energy dissipation rate, making that rate a natural adaptive signal.Controlling this product bounds the departure from physical energy, with a second-order step-size remainder.Source: §3.1 · pp. 6–7

3.2 Equidistribution framework

• Equidistributing g = |dE/dt| asymptotically minimizes the leading-order maximum symmetric-energy discrepancy for a fixed number of steps.The result assumes a positive dissipation-rate monitor; exactly flat intervals make the ideal monitor degenerate.Source: §3.2 · pp. 7–8
• A combined rule uses |dE/dt| during rapid dissipation and caps step growth near equilibrium, preserving the symmetric-energy discrepancy bound.Near equilibrium, the square-root-rate rule limits step growth even as the dissipation rate tends to zero; the combined rule's local discrepancy is bounded by the tolerance up to a second-order remainder.Source: §3.2 · pp. 8–9
• The scaled-arclength parameter β equals the ratio of the largest permitted step to the energy-error tolerance.Equivalently, 1/β is the dissipation rate at which the step-size and energy-error constraints balance.Source: §3.2 · p. 9

3.2.1 Energy–rate–based timestep selection

• The practical controller estimates the recent dissipation rate from the newest symmetric-energy difference and uses that estimate in the adaptive rules.The discrete estimator approximates the continuous rate to O(h), so the same bounds hold with modified constants.Source: §3.2.1 · p. 9

4.1 Algorithm

• The controller proposes a step from the energy-rate monitor, applies step and ratio bounds, then rejects and reduces steps until symmetric energy does not increase or the floor is reached.The rejection floor is max(Δtmin, ρminΔtn−1), preserving the lower step-ratio restriction needed for the convergence result.Source: §4.1 · p. 10

4.2 Energy stability of accepted steps

• Accepted symmetric-energy values form a monotone chain, yielding a physical-energy bound uniform in step count under the combined rule and stated smoothness assumptions.The tolerance caps the total admissible physical-energy rise over the integration, rather than only the rise per step.Source: §4.2 · p. 11
• The per-step physical-energy deviation depends on step-ratio changes and can persist as the step size tends to zero at a fixed ratio above one.When ratios are non-increasing, the middle term vanishes; when |ρn − 1| = O(Δtn), the bound is O(Δt_n^2) overall.Source: §4.2 · p. 11

4.3 Convergence

• SSAV–vBDF2 achieves second-order global accuracy for sufficiently smooth solutions when step sizes are bounded and ratios remain in the stated zero-stability range.The controller enforces bounded ratios, allowing standard variable-step convergence theory to apply.Source: §4.3 · p. 12

4.4 Time-stepper comparison

• For the Cahn–Hilliard comparison, Δt2 and the combined stepper perform best on energy error, followed by Δt1 and AM3.The comparison plots error against FFT count; Δt1 misses a small energy dip between t = 31 and 32, while Δt1 and scaled arclength behave similarly.Source: §4.4 · p. 12

5.1 Implementation details

• The implementation bounds physical step sizes between Δtmin and Δtmax = τ/δ and limits step-ratio changes using ρmin and ρmax.The examples set τ = τ1 = τ2, and use BDF1 for the first step.Source: §5.1 · pp. 12, 14

5.2 Convergence study with exact solution

• The Cahn–Hilliard convergence study shows second-order convergence for u and first-order convergence for max |Esym − E|.The reported errors average five runs with different random seeds.Source: §5.2 · p. 14, Table 2 · p. 15, Figure 3 · p. 15

5.3 Cahn–Hilliard spot collapse

• In the Cahn–Hilliard spot-collapse run, the time step decreases during the rapid energy drop around t = 19 and increases as energy flattens after t = 20.The experiment starts from two circles with different radii.Source: §5.3 · p. 15, Figure 4 · p. 16

5.4 Ohta–Kawasaki mixed states

• Across three Ohta–Kawasaki runs, time steps increase while energy is flat, decrease as it falls, then increase as it flattens; early rejections are negligible.The runs use different mean values, and rejected steps occur at the start.Source: §5.4 · p. 16, Table 3 · p. 16, Figure 5 · p. 17

5.5 Phase–Field Crystal in 3D

• For the 3D PFC run, step sizes fluctuate with the dissipation rate even when energy appears flat; 250 of 1,049 attempted steps are rejected, mostly within t ∈ [0, 2.5].At most one rejection occurs per time step.Source: §5.5 · p. 18, Figure 6 · p. 18, Figure 7 · p. 19

5.6 Functionalized Cahn–Hilliard

• In the FCH meandering-instability run, profiles resemble those in; time steps decrease as energy falls through t ∈ [0, 20] and increase after it flattens.The results are compared with.Source: §5.6 · pp. 18–19

5.7 Efficiency comparison

• Adaptive stepping gives approximately 10× speedup for Cahn–Hilliard and Ohta–Kawasaki, 18.9× for PFC, and 3.5× for FCH against matched-accuracy fixed steps.The comparison uses FFT evaluations and the L2 norm of |Esym − E(u)|.Source: §5.7 · p. 19

• g = |dot E| asymptotically minimizes the leading-order maximum symmetric-energy discrepancy for a fixed step budget, with optimum (E(0) - E(T))/(2N).Source: §6 · p. 19
• The combined monitor switches to sqrt(|dot E|) near equilibrium, capping step growth while keeping Delta t_n|dot E_n| bounded by tau1 in both regimes.Source: §6 · p. 19
• The prior-work arclength parameter beta is the ratio of the largest permitted step to the energy-error tolerance.Source: §6 · pp. 19, 22
• sup_n E(u^n) <= E(u^0) + tau1 + O(Delta t_max^2) bounds physical energy uniformly in step count after accepted symmetric-energy steps.The controller accepts steps only when the computable symmetric energy does not increase.Source: §6 · p. 22
• SSAV–vBDF2 achieves second-order global accuracy under standard step-ratio restrictions, and tests across four gradient-flow models show small steps during rapid energy changes and larger steps on plateaus.No accepted test step violated energy monotonicity.Source: §6 · p. 22
• Extensions to higher-order BDF methods, long-time bounds, and vector-valued flows require further analysis of truncation error, flow-specific asymptotics, and interacting dissipation mechanisms.Source: §6 · p. 22

• Writefull for Overleaf was used to paraphrase some sentences and make document edits.Source: Declaration of generative AI · p. 22
• Claude Code was used to comment on and debug some code.Source: Declaration of generative AI · p. 22
• The authors reviewed and edited the output and take full responsibility for the publication's content.Source: Declaration of generative AI · p. 22