AI 中文总结
针对二维周期受迫Navier–Stokes方程，提出IMEX-SDIRK2 mr-ccSAV方法，证明了无条件一致时间拟涡能等界，二阶最优收敛，实验验证其准确性、长时间鲁棒性及步长选择器有效性。
AI 中文摘要
我们提出并分析了一种针对涡量形式的二维周期受迫Navier--Stokes方程的IMEX-SDIRK2均值回复并行校正标量辅助变量（mr-ccSAV）方法。粘性项采用Alexander的SDIRK2方法处理，平流项采用显式处理。每个时间步需要求解两个带相同移位拉普拉斯算子的椭圆问题，以及一个三次或线性标量代数方程。当初始涡量属于$\u2d9f L^s(Ω)$（$s>2$）时，对任意正时间步长都存在阶段解；在显式小步长条件下可单独证明唯一性。核心结果是对任意正时间步长，直接得到无条件的一致时间拟涡能界。对于持续有界的外力，该估计是吸收性的：初始数据的影响会衰减，且外力的贡献不会随时间累积。在额外正则性、一致有界步长及有界相邻步长比的条件下，我们还在无小步长条件下证明了一致时间的$H^1$和$H^2$涡量界。对于光滑解，该方法以二阶精度最优收敛。数值实验验证了其准确性、长时间鲁棒性，以及配套嵌入式时间步长选择器的有效性。
英文摘要
We propose and analyze an IMEX-SDIRK2 mean-reverting concurrent-correction scalar auxiliary variable (mr-ccSAV) method for the forced two-dimensional periodic Navier--Stokes equations in vorticity form. The viscous term is treated by Alexander's SDIRK2 method and advection explicitly. Each stage requires two elliptic solves with the same shifted Laplacian and the solution of either a cubic or a linear scalar algebraic equation. For initial vorticity in $\dot L^s(Ω)$, $s>2$, a stage solution exists for every positive time step; uniqueness is established separately under an explicit small-step condition. The principal result is a direct, unconditional uniform-in-time enstrophy bound for arbitrary positive time steps. For persistently bounded forcing, this estimate is absorbing: the influence of the initial data decays, and the forcing contribution does not accumulate in time. Under additional regularity, uniformly bounded step sizes, and bounded neighboring step ratios, we also establish uniform-in-time $H^1$ and $H^2$ vorticity bounds without a small-step condition. For smooth solutions, the method converges optimally at second order. Numerical experiments confirm its accuracy, long-time robustness, and effectiveness of a companion embedded time-step selector.
