Finite differences for incompressible viscous fluid equations

I am working with the equations for incompressible viscous fluid:

$$\partial_t \vec{\omega} + (\vec{u}\cdot\nabla)\vec{\omega} = \nu\nabla^2\vec{\omega}$$ $$\nabla^2 \vec{\psi} = -\vec{\omega}$$ $$\vec{u}=\nabla \times \vec{\psi}$$

with an usual notation:

• $\omega$ ... vorticity
• $u$ ... velocity
• $\nu$ ... kinematic viscosity
• $\psi$ ... stream function

I need to discretize the system (by means of finite differences). Since this is very common tasks, the schemes must be somewhere available, but I was not lucky in googling this time. Could you please share a solution?

• How come? If I type "vorticity equation finite difference" into google, I get 233,000 results. See here: google.com/… – Wolfgang Bangerth Jun 3 '16 at 3:59
• One standard method is Chorin's method, also described on Wikipedia. It doesn't use the vorticity form, though. – David Ketcheson Jun 3 '16 at 10:58