Unconditionally Stable Second-Order Exponential Time-Differencing Schemes for the Incompressible NSE

We develop an efficient, unconditionally stable, variable-step second-order exponential time-differencing (ETD) scheme for the incompressible Navier–Stokes equations in two and three spatial dimensions under periodic boundary conditions, together with an embedded adaptive time-stepping variant. The scheme is unconditionally uniform-in-time stable in the sense that the numerical solution admits a time-uniform bound in $L^\infty(0,\infty;(L^2)^d)$ whenever the external forcing  term is uniformly bounded in time in $L^2$, for all Reynolds numbers and for arbitrary choices of time step sizes.At each time step, the method requires the solution of two time-dependent Stokes problems, which can be evaluated explicitly in the periodic setting using Fourier techniques, along with the solution of a single scalar cubic algebraic equation. Beyond the standard ETD framework, the proposed scheme incorporate two recently developed ingredients: (i) a dynamic second-order scalar auxiliary variable (SAV) correction, which is essential for achieving second-order temporal accuracy, and (ii) a mean-reverting scalar auxiliary variable (mr-SAV) multistep formulation, which plays a central role in ensuring long-time stability.The proposed methods overcome key limitations of existing approaches for the Navier–Stokes equations: classical Runge–Kutta schemes generally lack provable long-time stability, while IMEX and SAV-based BDF methods typically do not admit unconditional stability guarantees in the variable-step setting. Numerical experiments in two spatial dimensions confirm second-order temporal accuracy, uniform long-time stability, and effective error control provided by the adaptive strategy. Rigorous convergence analysis and a systematic investigation of long-time statistical properties will be pursued in future work.