Projection method

Definition

Consider the incompressible Navier–Stokes equations on a bounded domain $\Omega \subset \R^d$ with constant density $\rho$, kinematic viscosity $\nu$ and the no-slip boundary condition: $$\partial_t \mathbf{u} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho} \nabla p + \nu \Delta \mathbf{u}, \qquad \nabla \cdot \mathbf{u} = 0.$$ Let $\Delta t > 0$ be a time step and $\mathbf{u}^n$ the approximation of $\mathbf{u}$ at time $n \Delta t$. The projection method computes $\mathbf{u}^{n+1}$ and $p^{n+1}$ from $\mathbf{u}^n$ in three steps:

1. compute an intermediate velocity $\mathbf{u}^*$, equal to $0$ on $\partial\Omega$, from the momentum equation without pressure, for example $$\frac{\mathbf{u}^* - \mathbf{u}^n}{\Delta t} = -(\mathbf{u}^n \cdot \nabla) \mathbf{u}^n + \nu \Delta \mathbf{u}^*;$$
2. solve the Poisson equation $\Delta p^{n+1} = \frac{\rho}{\Delta t} \nabla \cdot \mathbf{u}^*$ in $\Omega$ with $\partial p^{n+1}/\partial \mathbf{n} = 0$ on $\partial \Omega$;
3. set $\mathbf{u}^{n+1} = \mathbf{u}^* - \frac{\Delta t}{\rho} \nabla p^{n+1}$.