Periodic advection schemes and the finite-interval upwind row

Use the index sets and uniform spacings of Uniform periodic, infinite, and Dirichlet grid conventions: $x_j=jh$, $j\in \mathbb{Z}$, or $j$ reduced modulo $N$ in the periodic case. Fix $a>0$, $h,k>0$, and $\nu =ak\mathrm{/}h$. The target equation is the one-dimensional constant-velocity transport equation $u_t+a{u}_x=0$, the constant-coefficient case of Linear transport equations and their characteristic flow. Set $U_j^0=u_0(jh)$ and define, for $n\ge 0$, $\begin{array}{rl}{\displaystyle \text{upwind: }}& {\displaystyle U_j^{n+1}=U_j^n-\nu (U_j^n-U_{j-1}^n),}\\ {\displaystyle \text{downwind: }}& {\displaystyle U_j^{n+1}=U_j^n-\nu (U_{j+1}^n-U_j^n),}\\ {\displaystyle \text{centered: }}& {\displaystyle U_j^{n+1}=U_j^n-\frac{\nu }{2}(U_{j+1}^n-U_{j-1}^n),}\\ {\displaystyle \text{Lax–Friedrichs: }}& {\displaystyle U_j^{n+1}=\frac{U_{j-1}^n+U_{j+1}^n}{2}-\frac{\nu }{2}(U_{j+1}^n-U_{j-1}^n),}\\ {\displaystyle \text{Lax–Wendroff: }}& {\displaystyle U_j^{n+1}=U_j^n-\frac{\nu }{2}(U_{j+1}^n-U_{j-1}^n)+\frac{{\nu }^2}{2}(U_{j+1}^n-2U_j^n+U_{j-1}^n)\mathrm{.}}\end{array}$ On a periodic grid, every index in these formulas is reduced modulo $N$; no physical boundary data are present. For $a<0$, reflect the stencil by $j\mapsto -j$ and replace $\nu$ by $\mathrm{\mid }a\mathrm{\mid }k\mathrm{/}h$.

For a finite interval $[0,L]$ with $a>0$, the inflow boundary is $x=0$. The upwind initial/boundary rows are $U_j^0=u_0(jh)$ for $j=0,\dots ,N$, $U_0^n=g(t_n)$ with $u_0(0)=g(0)$, and $U_j^{n+1}=(1-\nu )U_j^n+\nu U_{j-1}^n,j=1,\dots ,N,n\ge 0.$ The outflow node $j=N$ is computed by this row and receives no prescribed value. This item supplies no finite-interval boundary closure or stability statement for centered, Lax--Friedrichs, or Lax--Wendroff updates.