用于非定常纳维-斯托克斯方程的均值 informed 低秩整体随机伽辽金求解器
A Mean-Informed Low-Rank Monolithic Stochastic Galerkin Solver for the Unsteady Navier-Stokes Equations
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中文总结 AI 辅助
针对带不确定粘度的随机非定常不可压缩纳维-斯托克斯方程，提出一种均值 informed 的低秩整体随机伽辽金求解器，采用 TT 与 CP 格式表示解向量和系统矩阵，设计保留更多随机与非线性信息的预条件子，通过通道流实验验证其有效性。
中文摘要 AI 辅助
我们研究了带有不确定粘度的随机非定常不可压缩纳维-斯托克斯方程的低秩求解器。该问题采用随机伽辽金方法离散，并以整体（monolithic）形式表述，其中时间步长由均值问题的顺序求解提供信息。解向量以张量列车（Tensor Train, TT）格式表示，系统矩阵则根据需要以典型分解（CANDECOMP/PARAFAC, CP）格式表示。我们提出了基于低秩 CP 近似的预条件子，与基于均值的预条件子相比，其保留了额外的随机和非线性信息。针对基准通道流问题开展了数值实验，以验证低秩近似、相对容差策略及所提预条件子的有效性。
英文摘要
We study a low-rank solver for the stochastic unsteady incompressible Navier--Stokes equations with uncertain viscosity. The problem is discretized by a stochastic Galerkin method and written in an all-at-once (monolithic) form where the time steps are informed by a sequential solve of the mean problem. The solution vector is represented in Tensor Train (TT) format, while the system matrices are represented in CANDECOMP/PARAFAC (CP) format as needed. We propose preconditioners based on the low-rank CP approximations that retain additional stochastic and nonlinear information compared to mean-based preconditioners. Numerical experiments for a benchmark channel-flow problem are presented to show the effectiveness of the low-rank approximation, the relative tolerance strategy, and the proposed preconditioners.
