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 $g = |\dot{E}|$ 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 $g = \sqrt{|\dot{E}|}$ 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, $\sup_n E(u^n) \le E(u^{0}) + τ_1 + O(Δt_{\max}^2)$, 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.