纳维-斯托克斯流中精确移动边界使用浸入界面法的均匀误差估计
Uniform error estimates for Navier-Stokes flow with an exact moving boundary using the immersed interface method
AI总结：
本文研究低雷诺数纳维-斯托克斯流中精确移动边界问题，利用浸入界面法实现有限差分法的均匀二阶精度，证明速度误差为O(h²|log h|²)，压力误差为O(h²|log h|³)。
摘要：
我们证明在低雷诺数下，使用有限差分法处理纳维-斯托克斯流时，通过浸入界面法在界面附近将差分算子修正为O(h)精度，考虑速度梯度和压力的跃变，证明速度误差为均匀O(h²|log h|²)，压力误差为O(h²|log h|³)。证明使用了有限差分形式的泊松和扩散方程的估计，表现出最大范数下的正则性提升。
英文摘要：
We prove that uniform accuracy of almost second order can be achieved with a finite difference method applied to Navier-Stokes flow at low Reynolds number with a moving boundary, or interface, creating jumps in the velocity gradient and pressure. Difference operators are corrected to $O(h)$ near the interface using the immersed interface method, adding terms related to the jumps, on a regular grid with spacing $h$ and periodic boundary conditions. The force at the interface is assumed known within an error tolerance; errors in the interface location are not taken into account. The error in velocity is shown to be uniformly $O(h^2|\log{h}|^2)$, even at grid points near the interface, and, up to a constant, the pressure has error $O(h^2|\log{h}|^3)$. The proof uses estimates for finite difference versions of Poisson and diffusion equations which exhibit a gain in regularity in maximum norm.
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