One-dimensional orthogonal finite-volume viscous scheme

Let $L>0$, let $N\ge 1$ be an integer, put $h=L\mathrm{/}N$, and partition $\mathrm{\Omega }=(0,L)$ into the open cells $K_i=((i-1)h,ih)$, $i=1,\dots ,N$, with centers $x_i=(i-\frac{1}{2})h$. The interior face $F_{i+1\mathrm{/}2}=\{ih\}$ separates $K_i$ and $K_{i+1}$ for $i=1,\dots ,N-1$; the two endpoints are boundary faces. This is the one-dimensional uniform specialization of Finite-volume cell-face meshes and geometric regularity: cells are convex intervals, faces are their endpoints, and the common face orientation is from left to right.

For cell states $U_i\in \mathcal{G}$, set $v_i=v(U_i)$ and define the admissible face state $U_{i+1\mathrm{/}2}=(U_i+U_{i+1})\mathrm{/}2\in \mathcal{G}$. Define the face block $K_{i+1\mathrm{/}2}=K_{11}(U_{i+1\mathrm{/}2})$, where $K_{11}$ is the state-space block of Positive-semidefinite entropy-variable factorization of viscous flux for spatial direction one. The shared rightward viscous flux is

$G_{i+1\mathrm{/}2}=K_{i+1\mathrm{/}2}\frac{v_{i+1}-v_i}{h},i=1,\dots ,N-1.$

Set the boundary fluxes $G_{1\mathrm{/}2}=G_{N+1\mathrm{/}2}=0$. The finite-volume viscous operator is

${\dot{U}}_i=\frac{G_{i+1\mathrm{/}2}-G_{i-1\mathrm{/}2}}{h},i=1,\dots ,N\mathrm{.}$

This is only the viscous diffusion part of the balance law; no Euler convection term or time discretization is included. The boundary convention is zero total viscous flux.

The schematic shows $N=4$ cells and two representative shared-flux arrows. The arrows show the rightward orientation used in the balances; the endpoint labels are the imposed zero boundary fluxes, and the omitted interior face uses the same convention. The drawing illustrates signed incidences only and is not used as a proof.