An explicit jump immersed interface method for two-phase Navier–Stokes equations with interfaces

In this paper, we propose an explicit jump immersed interface method (EJIIM) for the incompressible Navier–Stokes equations with a discontinuous viscosity and singular forces along one or several interfaces in the solution domain. The EJIIM is used to get a second-order finite difference discretization at the grid points near or on the interface even if the jump conditions for the two-phase flow are complicated. The new method is based on a projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully second-order accuracy for the velocity and nearly second-order accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. The work here is a necessary first step in developing second-order accurate algorithm for two-phase Navier–Stokes equations with a moving interface.