# Stabilisation methods¶

When using a continuous Galerkin discretisation in advection-dominated problems, it may be necessary to stabilise the advection term in the momentum equation.

The implementation of the stabilisation methods can be found in the file stabilisation.py.

## Streamline upwind¶

This method adds some upwind diffusion in the direction of the streamlines. The term is given by

$\int_{\Omega} \frac{\bar{k}}{||\mathbf{u}||^2}(\mathbf{u}\cdot\nabla\mathbf{w})(\mathbf{u}\cdot\nabla\mathbf{u})$

which is added to the LHS of the momentum equation. The term $$\bar{k}$$ takes the form

$\bar{k} = \frac{1}{2}\left(\frac{1}{\tanh(\mathrm{Pe})} - \frac{1}{\mathrm{Pe}}\right)||\mathbf{u}||\Delta x$

where

$\mathrm{Pe} = \frac{||\mathbf{u}||\Delta x}{2\nu}$

is the Peclet number, and $$\Delta x$$ is the size of each element.