AI 中文总结
针对二维周期盒上的纳维 - 斯托克斯方程，提出线性变步长指数时间差分方法，结合ETD框架、mr - SAV及二阶外推，通过求解特定线性问题实现自适应时间步长与误差控制，证明了无条件长时间稳定性。
AI 中文摘要
我们针对二维周期盒上涡度 - 流函数形式的不可压缩纳维 - 斯托克斯方程提出了一种线性变步长指数时间差分方法。该方法由二阶格式和嵌入式一阶变体组成，为自适应时间步长和后验误差控制提供了自然机制。每个时间步仅需求解唯一可解的线性问题：两个热方程求解（在周期设置下通过傅里叶方法有效处理）和一个线性标量辅助变量方程（通过拉普拉斯变换和塔尔博特数值逆变换求值）。其构造结合了ETD框架、均值回复标量辅助变量（mr - SAV）和非线性项的二阶外推。均值回复校正实现了长时间稳定性并保持完全线性，这使其与需要非线性代数求解的相关mr - SAV格式不同。我们证明了无条件长时间稳定性：对于有界的\(L^2\)强迫，对于所有雷诺数和时间步长，离散涡度在\(L^\infty(0,\infty;L^2)\)中保持有界。进行了数值实验。
英文摘要
We propose a linear variable-step exponential time-differencing method for the incompressible Navier--Stokes equations in vorticity--streamfunction formulation on a two-dimensional periodic box. The method consists of a second-order scheme and an embedded first-order variant, yielding a natural mechanism for adaptive time stepping and a posteriori error control. Each time step requires only uniquely solvable linear problems: two heat equation solves, efficiently handled by Fourier methods in the periodic setting, and one linear scalar auxiliary-variable equation, evaluated via Laplace transform and Talbot's numerical inverse transform. The construction combines the ETD framework, a mean-reverting scalar auxiliary variable (mr-SAV), and second-order extrapolation of the nonlinear term. The mean-reverting correction enables long-time stability while preserving full linearity, distinguishing the method from related mr-SAV schemes that require nonlinear algebraic solves. We prove unconditional long-time stability: for uniformly bounded $L^2$ forcing, the discrete vorticity remains bounded in $L^\infty(0,\infty;L^2)$ for all Reynolds numbers and time-step sizes. Numerical experiments
