Convergence of a boundary integral method for 3-D water waves
Convergence of a boundary integral method for 3-D water waves
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DOI：
10.3934/dcdsb.2002.2.1
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发表时间：
2001-11
影响因子：
1.2
通讯作者：
T. Hou;Pingwen Zhang
中科院分区：
文献类型：
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作者：
T. Hou;Pingwen Zhang
证明了三维无粘无旋不可压缩流体中时变水波的改进点涡法的收敛性。我们的稳定性分析有两个重要组成部分。首先，我们推导出奇异速度积分的一阶近似。这个前阶近似捕获了原始速度积分对线性稳定性的所有前阶贡献。此外，该阶近似可以用Riesz变换表示，并且可以用谱精度进行近似。利用这一阶近似，我们构造了一个近场校正来稳定点涡法近似。经过近场校正后，改进的点涡方法线性稳定，并保持了连续速度积分的所有谱性质。通过在一致性误差中建立误差展开式，利用Strang技术获得了三阶精度的非线性稳定性和收敛性。
We prove convergence of a modified point vortex method for time-dependent water waves in a three-dimensional, inviscid, irrotational and incompressible fluid. Our stability analysis has two important ingredients. First we derive a leading order approximation of the singular velocity integral. This leading order approximation captures all the leading order contributions of the original velocity integral to linear stability. Moreover, the leading order approximation can be expressed in terms of the Riesz transform, and can be approximated with spectral accuracy. Using this leading order approximation, we construct a near field correction to stabilize the point vortex method approximation. With the near field correction, our modified point vortex method is linearly stable and preserves all the spectral properties of the continuous velocity integral to the leading order. Nonlinear stability and convergence with 3rd order accuracy are obtained using Strang’s technique by establishing an error expansion in the consistency error.