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

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 $g=\mathrm{\mid }\dot{E}\mathrm{\mid }$ 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 $g=\sqrt{\mathrm{\mid }\dot{E}\mathrm{\mid }}$ near equilibrium caps step growth without degrading this error bound. The arclength parameter $\beta$ 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, ${\sup }_{n}E(u^n)\le E(u^0)+{\tau }_1+O(\mathrm{\Delta }{t}_{\max }^{})$, 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.