Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 72 / 5 / b / Solution

For a sufficiently smooth function, Taylor expansion of the four neighboring values givesThe reaction term $\lambda u$ is sampled exactly at the grid point. On an exact solution, therefore,The method has second-order local accuracy for the partial differential equation. The original printed equations multiply this expression by $h^2$, so their unscaled residual is $O(h^4)$; that scaling does not make the Laplacian approximation fourth order. Resonance and inverse-matrix bounds are separate issues from this local truncation calculation.