一个用于非平稳纳维-斯托克斯方程的集合-正交分解方法
An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations
AI总结：
本文提出一种结合集合方法和正交分解的算法，用于高效求解非平稳纳维-斯托克斯方程多参数集解，提升计算效率与稳定性。
摘要：
偏微分方程（PDE）模型的定义通常涉及一组参数，其值可能在广泛范围内变化。即使单个参数集的解也可能非常昂贵。在许多情况下，例如优化、控制、不确定性量化等，需要多个参数集的解。我们考虑时间依赖的纳维-斯托克斯方程，其中一种 recently 开发的集合方法允许高效确定多个参数集对应的解。该方法利用任意时间步的多个解的平均值定义一组线性方程，以确定下一步的解。为了进一步降低确定纳维-斯托克斯方程多个解的成本，我们将在集合方法中引入正交分解（POD）降阶模型。集合方法的稳定性和收敛性结果被扩展到集合-POD方法。提供的数值实验展示了新方法在准确性和效率方面的表现。
英文摘要：
The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. To significantly further reduce the costs of determining multiple solutions of the Navier-Stokes equations, we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method. The stability and convergence results for the ensemble-based method are extended to the ensemble-POD approach. Numerical experiments are provided that illustrate the accuracy and efficiency of computations determined using the new approach.
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