Upwind SBP-SAT discretization of constant transport

Fix constant transport velocity $\beta \in {\mathbb{R}}^d$ and SBP data from Diagonal-norm summation-by-parts element data. For each cell $K$, let $U_K(t)$ be the vector of nodal values, $Q_{K,\mathrm{\ell }}=H_K{D}_{K,\mathrm{\ell }}$, and $U_{K,F}=R_{K,F}{U}_K$. Set $b_{K,F}=\beta \cdot n_{K,F}$. At each interior face, use the trace from the cell with positive outward speed when $b_{K,F}>0$, and the neighboring trace when $b_{K,F}<0$; at boundary faces use the inflow data on $b_{K,F}<0$ and the interior trace on $b_{K,F}\ge 0$. Denote this upwind scalar trace by ${\widehat{U}}_{K,F}$.

The upwind SBP-SAT semidiscretization is $H_K{\dot{U}}_K+{\sum }_{\mathrm{\ell }=1}^d{\beta }_{\mathrm{\ell }}{Q}_{K,\mathrm{\ell }}{U}_K+{\sum }_{F\subset \mathrm{\partial }K}{R}_{K,F}^T{M}_{K,F}{b}_{K,F}({\widehat{U}}_{K,F}-U_{K,F})=0.$ The sign of the SAT correction is fixed by this equation. The definition is spatial only and makes no fully discrete claim.