Backward difference operator ($D_h^{-}{W}_i=(W_i-W_{i-1})/h$)

The scaled backward finite difference uses the current and preceding nodal values to approximate a first derivative. If a function has two continuous derivatives, Taylor's theorem gives $D_h^{-}u(x)=u^{\prime }(x)-h{u}^{\prime \prime }(x)/2+o(h)$ as $h\to 0$. Composition with the scaled forward difference operator gives $D_h^{+}{D}_h^{-}u(x)=(u(x+h)-2u(x)+u(x-h))/h^2$, the second-order central difference for the second derivative.