DG formulation of the advection equation

Solutions obtained by the continous Galerkin (CG) formulation of PDEs dominated by advection or containing discontinuities typically suffer from numerical instabilities. A discontinuous Galerkin (DG) formulation provides a better alternative.

Strong form

See advection demo.

Weak form

\[\begin{split} \begin{aligned} &\text{Find}~u^{n+1}\in V~\text{such that} \\ &\begin{align*} F(u^{n+1}, v)&=\int_\Omega\text{d}\Omega~v\frac{u^{n+1} - u^n}{\Delta t^n} - \nabla\cdot(v\mathcal{D}_u(u)\mathcal{D}_{\textbf{a}}(\textbf{a})) \\ &\quad + \int_{\mathcal{F}}\text{d}\Gamma~\left[\!\left[ v\right]\!\right] f(\mathcal{D}_u(u)^+, \mathcal{D}_u(u)^-, \textbf{n}\cdot\textbf{a}) \\ &\quad + \int_{\partial\Omega_{\text{I}}}\text{d}\Gamma~vu_{\text{I}}\,\textbf{n}\cdot\mathcal{D}_{\textbf{a}}(\textbf{a}) \\ &\quad + \int_{\partial\Omega/\partial\Omega_{\text{I}}}\text{d}\Gamma~v\mathcal{D}_u(u)\,\textbf{n}\cdot\mathcal{D}_{\textbf{a}}(\textbf{a}) \\ &= 0 \quad\forall v\in V~. \end{align*} \\ &\text{where $f$ is a numerical flux between facets} \\ & f = \begin{cases} (\textbf{n}\cdot\textbf{a})u^+ & \text{if }\textbf{n}\cdot\textbf{a} > 0 \\ (\textbf{n}\cdot\textbf{a})u^- & \text{if } \textbf{n}\cdot\textbf{a} \leq 0 \\ \end{cases} \end{aligned} \end{split}\]