A Partially Implicit Hybrid Method for Computing Interface Motion in Stokes Flow

We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. We assume the interface is a closed curve which remains in the interior of the computational region. The implicit time integration is based on the small-scale decomposition approach and does not require the iterative solution of a system of nonlinear equations. First-order and second-order versions of the time-stepping method are derived systematically, and numerical results indicate that both methods are substantially more stable than explicit methods. At each time level, the Stokes equations are solved using a hybrid approach combining nearly singular integrals on a band of mesh points near the interface and a mesh-based solver. The solutions are second-order accurate in space and preserve the jump discontinuities across the interface. Finally, the hybrid method can be used as an alternative to adaptive mesh refinement to resolve boundary layers that are frequently present around a stiff immersed interface.