A numerical algorithm for viscous incompressible interfacial flows

We present a new algorithm to numerically simulate two-dimensional viscous incompressible flows with moving interfaces. The motion is updated in time by using the backward difference formula through an iterative procedure. At each iteration, the pseudo-spectral technique is applied in the horizontal direction. The resulting semi-discretized equations constitute a boundary value problem in the vertical coordinate which is solved by decoupling growing and decaying solutions. Numerical tests justify that this method achieves fully second-order accuracy in both the temporal variable and vertical coordinate. As an application of this algorithm, we study the motion of Stokes waves in the presence of viscosity. Our numerical results are consistent with the recently published asymptotic solution for Stokes waves in slightly viscous fluids.