The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

Peskinʹs Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be first-order accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a second-order 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. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper.