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.