切割单元有限体积算子中的缺陷子空间与局域不稳定性
Spectral Analysis and Redistribution Thresholds for Cut-Cell Finite-Volume Methods
AI总结：
本文研究切割单元有限体积算子的缺陷子空间与局域不稳定性，推导可局部计算的SRD混合参数判据，表征相关算子结构，防止需重跑的模拟崩溃。
AI中文摘要：
显式有限体积方法中的切割单元不稳定性产生于嵌入边界将背景单元切割为小体积分数α时，迫使局部CFL数λ_α=λ/α≫1。本文表征了该不稳定性的谱结构，并推导了每个切割单元可通过局部网格几何计算的最小SRD混合参数。更新算子恰好有m个不稳定特征值，每个切割单元对应一个，主导特征值为μ_u=1−λ/α+O(1)，特征向量为v_u=e_c+O(α)。增长率由缺陷矩阵Γ=(Δt/h)(a_{c+1/2,c}−a_{c−1/2,c})控制，得到判据s_stab=(λ_α−2)/(λ_α−1)+O(α)，该判据在网格生成时通过α和λ计算，无需时间步长和全局矩阵组装。此判据可防止需以高昂成本重跑生产模拟的崩溃类问题。对于左合并状态重分布，|cos_α(D,v_u)|=1−O(α)，证明SRD作用于不稳定特征方向。本文还表征了标量周期一维切割单元网格上的哪些线性化显式有限体积算子具有该结构，涵盖从一阶Godunov到带有界通量系数的高阶MUSCL格式，并证明对于m个非相邻切割单元，||P_{𝒰_α}−P_𝒞||=O(α)。
英文摘要：
In finite-volume methods on embedded-boundary meshes, arbitrarily small cut cells can produce coefficients that scale as O(alpha^{-1}) when the time step is chosen for the regular grid. For a one-dimensional periodic upwind discretization, we show that the resulting instability is carried by an eigenmode concentrated at the cut cell. Blending the unstabilized update with the volume-weighted state obtained by merging the cut cell with its left neighbor yields the leading redistribution threshold s_0(L)=(L-2)/(L-1) for fixed cut-cell Courant number L>2. We prove that the cell-merging correction aligns with the unstable mode as alpha tends to zero and establish a block-matrix result showing that diverging cut-cell rows generate a finite cluster of unbounded eigenvalues whose invariant subspace approaches the span of the cut-cell coordinates. For MUSCL with a minmod limiter, analysis of the piecewise-linear region containing the localized mode gives s_M(alpha,L)=(L-2)/(L-1)-L(L-2)alpha/[4(L-1)^2]+O(alpha^2), with the SSPRK2 update reaching unit amplification at the same crossing on that branch. Numerical tests show lower error with this reduced redistribution than with the tested published weighting specializations and full two-cell merging. A two-dimensional 45-degree embedded-boundary channel likewise reaches spectral radius at most one with blending parameters below full merging. These results show that full redistribution is not always necessary to control the unstable cut-cell mode, and that using a smaller blending parameter can reduce numerical error in the tested problems.
