by: find $u_h\in V_h$ such that for all $v_h\in V_h$ following
equation holds:
\begin{equation}\label{dg-transport}
- \sum_\kappa\left\{-(u_h,\boldsymbol\beta\cdot\nabla v_h)_\kappa+(\boldsymbol\beta\cdot{\bf n}\, u_h, v_h)_{\partial\kappa_+\setminus\Gamma}+(\boldsymbol\beta\cdot{\bf n}\, u_h^-, v_h)_{\partial\kappa_-\setminus\Gamma}\right\}=(f,v_h)_\Omega-(\boldsymbol\beta\cdot{\bf n}\, g, v_h)_{\Gamma_-},
+ \sum_\kappa\left\{-(u_h,\boldsymbol\beta\cdot\nabla v_h)_\kappa
+ +(\boldsymbol\beta\cdot{\bf n}\, u_h, v_h)_{\partial\kappa_+}
+ +(\boldsymbol\beta\cdot{\bf n}\, u_h^-, v_h)_{\partial\kappa_-\setminus\Gamma}\right\}
+ =(f,v_h)_\Omega-(\boldsymbol\beta\cdot{\bf n}\, g, v_h)_{\Gamma_-},
\end{equation}
where $\partial\kappa_-:=\{x\in\partial\kappa,
\boldsymbol\beta(x)\cdot{\bf n}(x)<0\}$ denotes the inflow boundary
to as \emph{first version} of the DG method. We note that after a
second integration by parts, we obtain: find $u_h\in V_h$ such that
\[
- \sum_\kappa\left\{(\nabla\cdot\{\boldsymbol\beta u_h\},\nabla v_h)_\kappa-(\boldsymbol\beta\cdot{\bf n} [u_h], v_h)_{\partial\kappa_-}\right\}=(f,v_h)_\Omega, \quad\forall v_h\in V_h,
+ \sum_\kappa\left\{(\nabla\cdot\{\boldsymbol\beta u_h\},v_h)_\kappa
+ -(\boldsymbol\beta\cdot{\bf n} [u_h], v_h)_{\partial\kappa_-}\right\}
+ =(f,v_h)_\Omega, \quad\forall v_h\in V_h,
\]
where $[u_h]=u_h^+-u_h^-$ denotes the jump of the discrete function
between two neighboring cells and is defined to be $[u_h]=u_h^+-g$ on