 Midpoint flux stencil for one-dimensional diffusion

For $u_t=(a{u}_x{)}_x$, this conservative finite difference method uses coefficient samples at edge midpoints. For smooth $a,u$, its spatial residual is $h^2(a{u}_{xxxx}/12+a^{\prime }{u}_{xxx}/6+a^{\prime \prime }{u}_{xx}/8+a^{\prime \prime \prime }{u}_x/24)+O(h^4)$. With zero Dirichlet boundary conditions, $-v^T{L}_{h}v=h^{-2}{\sum }_j{a}_{j+1/2}(v_{j+1}-v_j{)}^2$. If $0<a\le a_{+}$, its eigenvalues lie in $[-4a_{+}/h^2,0]$, giving the mesh-independent forward Euler method bound $\Delta t/h^2\le 1/(2a_{+})$. Constant $a=a_{+}$ proves sharpness as the grid is refined.