A family of second-order time stepping methods for a nonlinear fluid-fluid interaction model

In this paper, we present a fully discrete finite element scheme for the nonlinear fluid-fluid interaction model, which consists of two Navier-Stokes equations coupled by some nonlinear interface. The presented fully discrete scheme is based on a type of implicit-explicit (IMEX) second-order time-stepping schemes in temporal discretization and mixed finite element in spatial discretization. The scheme is a combination of a linearization treatment for the advection term, explicit treatment for nonlinear interface conditions by a scalar auxiliary variable method, together with stabilization terms which are proportional to discrete curvature of the solutions in both velocity and pressure. Because of the scalar auxiliary variable method, we only require solving a sequence of linear differential equation with constant coefficients at each time step. Unconditional stability is proved and convergence analysis is derived. Finally, the derived theoretical results are supported by three numerical examples.