Energy-rate equidistribution for adaptive SSAV--BDF2 integration of gradient flows
Abstract
We develop an adaptive time-stepping strategy for scalar auxiliary variable (SAV) schemes combined with variable-step BDF2 discretization for gradient flows. The strategy selects step sizes based on the instantaneous energy dissipation rate and enforces monotonicity of a computable symmetric energy through a step rejection mechanism. We show that the step-size rules used in practice correspond to equidistributing a monitor function along the energy curve, connecting energy-adaptive time stepping to the classical theory of r-adaptive mesh methods. The monitor g = |Ė| is asymptotically optimal as it minimises the leading-order maximum symmetric-energy discrepancy for a given number of steps. A combined monitor that switches to g = √(|Ė|) near equilibrium caps step growth without degrading this error bound. The arclength parameter β used in prior work is shown to be the ratio of the largest permitted step to the energy-error tolerance. The controller is paired with the SSAV--BDF2 discretization. Accepted steps satisfy monotonicity of the symmetric energy, from which we deduce a bound on the physical energy that is uniform in the number of steps, supn E(un) ≤ E(u0) + τ1 + O(Δtmax2), and the scheme achieves second-order global accuracy under standard step-ratio restrictions. Numerical experiments on Cahn--Hilliard, Ohta--Kawasaki, Phase--Field Crystal, and functionalized Cahn--Hilliard models confirm the theory and demonstrate the method's efficiency.
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BibTeX
@misc{shergill2026energyrate,
title = {{Energy-rate equidistribution for adaptive SSAV--BDF2 integration of gradient flows}},
author = {Suki Shergill and JF Williams},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30454},
eprint = {2610.30454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30454},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Energy-rate equidistribution for adaptive SSAV--BDF2 integration of gradient flows
AU - Suki Shergill
AU - JF Williams
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30454
UR - https://arxiv.org/abs/2610.30454
AB - We develop an adaptive time-stepping strategy for scalar auxiliary variable (SAV) schemes combined with variable-step BDF2 discretization for gradient flows. The strategy selects step sizes based on the instantaneous energy dissipation rate and enforces monotonicity of a computable symmetric energy through a step rejection mechanism. We show that the step-size rules used in practice correspond to equidistributing a monitor function along the energy curve, connecting energy-adaptive time stepping to the classical theory of r-adaptive mesh methods. The monitor g = |Ė| is asymptotically optimal as it minimises the leading-order maximum symmetric-energy discrepancy for a given number of steps. A combined monitor that switches to g = √(|Ė|) near equilibrium caps step growth without degrading this error bound. The arclength parameter β used in prior work is shown to be the ratio of the largest permitted step to the energy-error tolerance. The controller is paired with the SSAV--BDF2 discretization. Accepted steps satisfy monotonicity of the symmetric energy, from which we deduce a bound on the physical energy that is uniform in the number of steps, sup_n E(u^n) ≤ E(u^0) + τ_1 + O(Δt_max^2), and the scheme achieves second-order global accuracy under standard step-ratio restrictions. Numerical experiments on Cahn--Hilliard, Ohta--Kawasaki, Phase--Field Crystal, and functionalized Cahn--Hilliard models confirm the theory and demonstrate the method's efficiency.
ER -