Four-point midpoint third derivative ($\frac{u_{j+2}-3u_{j+1}+3u_j-u_{j-1}}{h^3}=u^{\prime \prime \prime }(x_{j+1/2})+O(h^2)$)

The antisymmetric four-node stencil at offsets $-3h/2,-h/2,h/2,3h/2$ about a midpoint approximates the third derivative. Taylor expansion gives leading error $h^2{u}^{(5)}/8$. For a third-order boundary value problem on nodes $0,\dots ,n+1$, use rows $j=1,\dots ,n-1$, prescribed endpoint values and one consistent one-sided derivative condition. This avoids a ghost node while providing the correct number of equations. For instance a known right-endpoint first derivative is approximated by $(3U_{n+1}-4U_n+U_{n-1})/(2h)$.