A Fourier spectral immersed boundary method with exact translation invariance, improved boundary resolution, and a divergence-free velocity field

Abstract: 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.