Upwind discontinuous Galerkin discretization of constant transport

Let $d\ge 1$, let $\beta \in {\mathbb{R}}^d$ be constant, and consider $u_t+\mathrm{\nabla }\cdot (\beta u)=0$ on a fixed domain. For $d=1$, use Oriented interval meshes and broken polynomial spaces. For $d\ge 2$, use a finite matching affine simplicial mesh ${\mathcal{T}}_h$ and its broken space $V_h^{\mathrm{b}\mathrm{r}}={\prod }_{K\in {\mathcal{T}}_h}{\mathbb{P}}_p(K)$ from Matching affine simplicial meshes and shape regularity and Broken and conforming polynomial spaces on a matching mesh. On each cell $K$, let $u_h{\mathrm{\mid }}_K\in {\mathbb{P}}_p(K)$, with $p\ge 0$.

In this discretization, a cell face means a facet of dimension $d-1$ when $d\ge 2$, and an endpoint when $d=1$; all face sums below use only these faces, with counting measure at endpoints. For each cell face $F\subset \mathrm{\partial }K$, let $n_{K,F}$ be the outward unit normal, $b_{K,F}=\beta \cdot n_{K,F}$, and $u_K$ the trace of $u_h{\mathrm{\mid }}_K$. On an interior face with neighbor $K^{\mathrm{\prime }}$, define the upwind trace by ${\widehat{u}}_{K,F}=u_K$ when $b_{K,F}>0$ and ${\widehat{u}}_{K,F}=u_{K^{\mathrm{\prime }}}$ when $b_{K,F}<0$; either trace may be used when $b_{K,F}=0$. On a physical boundary face, use the prescribed inflow datum $g$ when $b_{K,F}<0$ and $u_K$ when $b_{K,F}\ge 0$. The numerical flux is ${\widehat{f}}_{K,F}=b_{K,F}{\widehat{u}}_{K,F}$.

The upwind DG semidiscretization seeks a differentiable cellwise polynomial $u_h(t)$ such that for every $K\in {\mathcal{T}}_h$ and $v_h\in {\mathbb{P}}_p(K)$, ${\int }_K({\mathrm{\partial }}_t{u}_h)v_{h}dx-{\int }_K\beta u_h\cdot \mathrm{\nabla }{v}_{h}dx+{\sum }_{F\subset \mathrm{\partial }K}{\int }_F{\widehat{f}}_{K,F}{v}_{h}dS=0.$ This definition is spatial only. It asserts neither a fully discrete bound nor a shock-convergence result.

For the face orientation, in the diagram $F$ is the shared vertical edge, $n_F=(1,0)$ points from $K^{-}$ to $K^{+}$, and $b=\beta \cdot n_F$. The flux takes the trace on the side from which the sign of $b$ transports data. The coordinates and orientation specify the diagram completely; the proofs use the displayed flux formula, not the picture.