Navier-Stokes方程变分初始状态数据同化的Reynolds半稳健、全局无散E-HDG/IMEX-SAV方法

中文摘要 AI 辅助

本文针对由非定常不可压缩Navier-Stokes方程控制的变分初始状态数据同化问题，提出了一种嵌入式混合化间断伽辽金（E-HDG）方法，并结合一阶隐式-显式标量辅助变量（IMEX-SAV）时间离散格式。我们采用先优化后离散的策略。空间离散分别使用次数为k和k-1的不连续分段多项式表示单元速度和压力，连续次数为k的速度迹，以及不连续次数为k的压力迹。由此得到的状态和伴随速度以及重构的初始速度均全局无散。前向IMEX-SAV状态格式无条件能量稳定。在适当的正则性和局部轨迹假设（其界限不显式包含粘度的负幂）、网格相关条件Δt≲h²以及Tikhonov参数的收缩性条件下，我们建立了全离散OTD最优性系统的局部存在唯一性，并给出了状态、伴随和重构初始速度的阶为O(h^k+Δt)的Reynolds半稳健L²误差估计；常数不显式包含粘度的负幂。数值实验在光滑测试中确认了收敛性，并展示了机器精度的离散不可压缩性、有效的非线性求解器行为以及在低粘度区域中的稳定性能。

英文摘要

This paper develops an embedded-hybridized discontinuous Galerkin (E-HDG) method combined with a first-order implicit-explicit scalar auxiliary variable (IMEX-SAV) time discretization for variational initial-state data assimilation governed by the unsteady incompressible Navier-Stokes equations. We adopt an optimize-then-discretize strategy. The spatial discretization uses discontinuous piecewise polynomials of degrees $k$ and $k-1$ for the element velocity and pressure, respectively, a continuous degree-$k$ velocity trace, and a discontinuous degree-$k$ pressure trace. The resulting state and adjoint velocities, as well as the reconstructed initial velocity, are globally divergence-free. The forward IMEX-SAV state scheme is unconditionally energy stable. Under suitable regularity and local-trajectory assumptions whose bounds introduce no explicit negative powers of the viscosity, the mesh-dependent condition $Δt\lesssim h^2$, and a contractivity condition on the Tikhonov parameter, we establish local existence and uniqueness of the fully discrete OTD optimality system and Reynolds-semi-robust $L^2$ error estimates of order $O(h^k+Δt)$ for the state, adjoint and reconstructed initial velocities; the constants contain no explicit negative powers of the viscosity. Numerical experiments confirm convergence on smooth tests and demonstrate machine-precision discrete incompressibility, effective nonlinear-solver behavior, and stable performance in small-viscosity regimes.

发表机构

• School of Mathematics, Sichuan University（四川大学数学学院）

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