Forward-Euler step for degree-zero upwind DG transport

Input a dimension $d\ge 2$, a finite matching affine simplicial mesh ${\mathcal{T}}_h$ whose boundary faces are paired by translations to form a periodic mesh, a constant velocity $\beta \in {\mathbb{R}}^d$, one cell mean $U_K\in \mathbb{R}$ per cell, and a step $\mathrm{\Delta }t>0$. The output is either CFL-inadmissible or an updated vector of cell means $U^{+}=(U_K^{+}{)}_{K\in {\mathcal{T}}_h}$. Let $V_K={\int }_{K}1dx$ be the cell volume and $A_F={\int }_{F}1dS$ the face area. Here a cell face is a codimension-one facet, as in Upwind discontinuous Galerkin discretization of constant transport. For each face $F\subset \mathrm{\partial }K$, let $n_{K,F}$ be the outward unit normal, $b_{K,F}=\beta \cdot n_{K,F}$, and $K_F$ the neighboring cell across that periodic face. With $b^{+}=\max (b,0)$ and $b^{-}=\min (b,0)$, the upwind flux out of $K$ is ${\widehat{f}}_{K,F}=b_{K,F}^{+}{U}_K+b_{K,F}^{-}{U}_{K_F}\mathrm{.}$

Compute ${\lambda }_K=\frac{\mathrm{\Delta }t}{V_K}{\sum }_{F\subset \mathrm{\partial }K}{A}_F{b}_{K,F}^{+}\mathrm{.}$ If ${\lambda }_K>1$ for any cell, return CFL-inadmissible without updating the state. Otherwise, the degree-zero upwind forward-Euler update is admissible and its output is $U_K^{+}=U_K-\frac{\mathrm{\Delta }t}{V_K}{\sum }_{F\subset \mathrm{\partial }K}{A}_F{\widehat{f}}_{K,F}\mathrm{.}$ This is the degree-zero specialization of Upwind discontinuous Galerkin discretization of constant transport.