体积滤波浸入边界法中强迫项与亚滤波尺度项的表征
Characterization of the forcing and sub-filter scale terms in the volume-filtering immersed boundary method
AI总结：
本文针对体积滤波浸入边界法中的强迫项与亚滤波尺度项，基于二维变系数双曲方程开展数值分析，提出双曲PDE体积滤波的系统方法，验证了该方法的二阶收敛性及亚滤波尺度项与滤波尺寸平方的比例关系。
摘要：
本文对Dave等人2023年发表于《计算物理杂志》（JCP）的体积滤波浸入边界（VF-IB）方法中产生的强迫项与亚滤波尺度项进行了表征。体积滤波过程会产生面积分形式的体积力，用以描述界面处的边界条件。此外，该方法还会产生未封闭的亚滤波尺度（SFS）项。SFS项对数值解的贡献程度取决于滤波宽度。为了更好地理解这些项，我们采用了Brady与Liverscu2021年发表于《计算物理杂志》的二维变系数双曲方程。选择该算例有两个原因：首先，该算例包含被界面分隔的两个不同区域，是VF-IB方法的理想测试算例；其次，已有的解析解使我们能够针对不同滤波尺寸恰当研究SFS项的贡献，而滤波尺寸决定了界面的分辨程度——滤波尺寸越小，界面的分辨程度越高。本文对该方法开展了全面的数值分析，并研究了SFS项对数值解的影响。为进行直接对比，我们将数值解与滤波后的解析解做了比较。通过该研究我们明确了三个重要结论：第一，提出了一种对双曲偏微分方程（PDE）进行体积滤波的系统性方法；第二，证明了VF-IB方法在滤波尺寸减小（即界面更锐利）时具有二阶收敛性；第三，证明了SFS项与滤波尺寸的平方成正比。大滤波尺寸下需要对SFS项进行建模，但对于足够精细的滤波器，忽略SFS项不会导致解的精度出现显著下降。
英文摘要：
We present a characterization of the forcing and the sub-filter scale terms produced in the volume-filtering immersed boundary (VF-IB) method by Dave et al, JCP, 2023. The process of volume-filtering produces bodyforces in the form of surface integrals to describe the boundary conditions at the interface. Furthermore, the approach also produces unclosed subfilter scale (SFS) terms. The level of contribution from SFS terms on the numerical solution depends on the filter width. In order to understand these terms better, we take a 2 dimensional, varying coefficient hyperbolic equation shown by Brady & Liverscu, JCP, 2021. This case is chosen for two reasons. First, the case involves 2 distinct regions seperated by an interface, making it an ideal case for the VF-IB method. Second, an existing analytical solution allows us to properly investigate the contribution from SFS term for varying filter sizes. The latter controls how well resolved the interface is. The smaller the filter size, the more resolved the interface will be. A thorough numerical analysis of the method is presented, as well as the effect of the SFS term on the numerical solution. In order to perform a direct comparison, the numerical solution is compared to the filtered analytical solution. Through this, we highlight three important points. First, we present a methodical approach to volume filtering a hyperbolic PDE. Second, we show that the VF-IB method exhibits second order convergence with respect to decreasing filter size (i.e. making the interface sharper). Finally, we show that the SFS term scales with square the filter size. Large filter sizes require modeling the SFS term. However, for sufficiently finer filters, the SFS term can be ignored without any significant reduction in the accuracy of solution.
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