This paper introduces a new immersed boundary (IB) method for viscous incompressible flow, based on a Fourier spectral method for the fluid solver and on the nonuniform fast Fourier transform (NUFFT) algorithm for coupling the fluid with the immersed boundary. The new Fourier spectral immersed boundary (FSIB) method gives improved boundary resolution in comparison to the standard IB method. The interpolated velocity field, in which the boundary moves, is analytically divergence-free. The FSIB method is gridless and has the meritorious properties of volume conservation, exact translation invariance, conservation of momentum, and conservation of energy. We verify these advantages of the FSIB method numerically both for the Stokes equations and for the Navier-Stokes equations in both two and three space dimensions. The FSIB method converges faster than the IB method. In particular, we observe second-order convergence in various problems for the Navier-Stokes equations in three dimensions. The FSIB method is also computationally efficient with complexity of $O(N^3\log(N))$ per time step for $N^3$ Fourier modes in three dimensions.

翻译：本文提出了一种新的用于粘性不可压缩流动的浸入边界（IB）方法，该方法基于流体求解器的傅里叶谱方法以及用于流体与浸入边界耦合的非均匀快速傅里叶变换（NUFFT）算法。与标准IB方法相比，新的傅里叶谱浸入边界（FSIB）方法提供了改进的边界分辨率。边界在其中移动的插值速度场在解析上是无散的。FSIB方法是无网格的，并具有体积守恒、精确平移不变性、动量守恒和能量守恒等优良性质。我们通过数值验证了FSIB方法在斯托克斯方程和纳维-斯托克斯方程（分别在二维和三维空间）中的这些优势。FSIB方法的收敛速度优于IB方法。特别地，我们在三维纳维-斯托克斯方程的各种问题中观察到了二阶收敛。FSIB方法在计算上也是高效的，对于三维空间中的$N^3$傅里叶模态，每时间步的计算复杂度为$O(N^3\log(N))$。