A third-order Immersed Interface Method for the velocity-pressure Navier-Stokes equations on collocated grids

Combining a finite-difference discretization with an immersed interface method (IIM) has been shown to yield high-order spatial accuracy for solving simple PDEs on complex domains. Representing the quantities on collocated grids enables both a third-order or higher discretization error, but also retains the simplicity and scalability associated with explicit finite-difference stencils. Extending this IIM approach to the incompressible Navier-Stokes equations while achieving high order spatial and temporal accuracy is an open challenge. In this talk we present a third-order Runge-Kutta based algorithm to solve the incompressible Navier-Stokes equations using a third-order IIM boundary discretization on collocated grids. We present and analyze convergence and stability results for flows with stationary and moving boundaries in 2D, though the algorithm should readily translate to 3D as well.