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

TL;DR

This work addresses accurate and stable long-time integration of the incompressible NSE under periodic boundaries by introducing unconditionally stable, variable-step, second-order ETD schemes that preserve uniform-in-time energy bounds. Central to the approach is a mean-reverting scalar auxiliary variable (mr-SAV) combined with a dynamic second-order SAV correction and ETD multistep discretization, enabling fully linear solves in the fluid variables. The authors establish unconditional long-time stability and demonstrate second-order temporal accuracy, along with an embedded adaptive scheme that automatically controls error and improves efficiency. Numerical experiments in 2D confirm convergence, stability, and statistically consistent long-time behavior, with adaptive stepping yielding substantial computational savings. The framework promises robust integration for long-time simulations and offers potential extensions to non-periodic domains, higher-order schemes, and data-driven modeling contexts.

Abstract

We develop an efficient, unconditionally stable, variable step second order exponential time differencing 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 Linfinity over time with values in L2 to the power d whenever the external forcing term is uniformly bounded in time in L2, 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 exponential time differencing framework, the proposed scheme incorporates two recently developed ingredients. The first is a dynamic second order scalar auxiliary variable correction, which is essential for achieving second order temporal accuracy. The second is a mean reverting scalar auxiliary variable 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 scalar auxiliary variable 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.

Figures (7)

• Figure 1: The $L^2$ error of velocity (left), vorticity (middle), and absolute error of the auxiliary variable (right), computed by ETD-mr-SAV-MS1o and ETD-mr-SAV-MS2o at $T=1$, plotted against the time-step size $\tau = 0.1\times 2^{-k}$, for $k = 0,1,\cdots,6$.
• Figure 2: The $L^2$ error of velocity (left), vorticity (middle), and absolute error of the auxiliary variable (right), computed by ETD-mr-SAV-MS1o and ETD-mr-SAV-MS2o at $T=1$, plotted against the number of time steps $N=2^k$ under a $10\%$ perturbed variable-step sequence, for $k=5,\dots,10$.
• Figure 3: Evolution of the vorticity $L^2$ norm computed by the ETD-mr-SAV-MS2o and ETD-mr-SAV-MS12 schemes for $\nu=1/20$ and $m=2$.
• Figure 4: Enstrophy ($L^2$ norm of vorticity) versus time computed by ETD-mr-SAV-MS2o and ETD-mr-SAV-MS12 for $\nu=1/40$ and $m=4$. The red dotted line denotes the mean value $\mu$; the orange dashed and green dash-dotted lines correspond to the thresholds $\mu+2\sigma$ and $\mu+3\sigma$, respectively.
• Figure 5: (a) Adaptive time step sizes as a function of time. (b) Number of steps over time for constant steps and adaptive steps. (c) Absolute value of the auxiliary variable as a function of time. (d) Relative $L^2$ error indicator of vorticity based on numerical solutions and (e) Reference relative $L^2$ error based on a refined ETDRK4 solution.

Theorems & Definitions (11)

• Remark 3.1: Solvability of the auxiliary variable
• Remark 3.2: Second-order accuracy
• Remark 3.3: Alternative second-order schemes
• Remark 3.4
• Lemma 4.1
• Lemma 4.2
• Theorem 4.3
• proof
• Remark 4.1
• Example 5.1