Partially implicit motion of a sharp interface in Navier-Stokes flow

We develop a numerical method for the coupled motion of Navier-Stokes flow with an elastic interface of zero thickness which exerts tension and bending forces on the fluid. The interface motion is made partially implicit by approximating a backward Euler step in the high wavenumbers as in the small scale decomposition method of Hou, Lowengrub and Shelley. This modified step is combined with the method of [J. T. Beale, A. T. Layton, A velocity decomposition approach for moving interfaces in viscous fluids, J. Comput. Phys. 228 (2009) 3358–67]; the fluid velocity is found by computing the Stokes velocity and a more regular remainder. The resulting scheme is second order in space and first order in time; it can be made second order in time by extrapolation. The discontinuities in the pressure and velocity gradient are preserved. The partially implicit method allows much larger time steps than an explicit method with negligible added effort. The formulas in the Fourier transform for the implicit approximation in high wavenumbers are similar to those derived in [T. Y. Hou, Z. Shi, An efficient semi-implicit immersed boundary method for the Navier-Stokes equations, J. Comput. Phys. 227 (2008) 9138–69] in a different context.