Hi,
I am trying to solve a non-linear purely convective PDE of the form:
$$ \frac{\partial{z}}{\partial{t}} + \nabla{Q} = 0 $$
Where, dz/dt is the partial time derivative of the unknown variable and Q is a flux vector, which is a non-linear function and it can be computed independently at the node centres.
I am trying to see if is possible to resolve this PDE with a DG approximation so the discretised equation should look like this:
$$ \int_{\Omega_{e}}\frac{\partial{z_h}}{\partial{t}}v_h d\Omega_{e} - \int_{\Omega_{e}}Q_h\cdot\nabla{v_h} d\Omega_{e} + \int_{\Gamma_{e}}\hat{Q_h}v_hd\Gamma_{e} = 0 $$
where z is the solution, and Q^ is the numerical flux which could be computed from the two fluxes Q- and Q+ on the adjacent elements with an upwind scheme for example.
At the end I would like to resolve an ODE system like:
$$ M\frac{\partial{z^{i}}}{\partial{t}} - KQ^{i} + M^{+}\hat{Q^{+}} - M^{-}\hat{Q^{-}} = 0 $$
So my question is as follows: Is it possible to use the UFL to write the forms in a way that the solver could compile for this kind of system or am I missing something else?
Is there an existing demo or any work with similar equations or methods using DOLFIN?
Thanks,
J
