Abstract

In this course we consider projection methods for the numerical approximation of time-dependent incompressible fluid equations. Such schemes are based on the projection structure of the equations due to the incompressibility constraint. An operator splitting leads to a prediction and a correction step in each time step both of which are simpler problems. For this reason (high order) projection methods are of particular interest for large scale simulations. But on the other hand the splitting introduces additional challenges e.g. regarding the boundary conditions.

We start by reviewing projection methods for viscous flows (Navier-Stokes and Stokes equations) dating back to Chorin and Temam and compare them to mixed methods. Then we take a look at what has been done for inviscous equations (Euler). Finally we present the Green-Naghdi model that is derived from the incompressible free surface Euler equations. We discuss how its projection structure helps to pose meaningful boundary conditions.