Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 39B (Numerical Analysis)

Expand the two central second-difference operators. The given stencil identity yieldsConsequently its unnormalized local residual is ${\Gamma }_{9}u-h^{2}f=(h^4/12)\Delta f+O(h^6)$. The normalized Laplacian approximation has leading error $(h^2/12)\Delta f$ and is second order. These Taylor statements require the indicated smooth derivatives; boundary accuracy is a separate matter.

If $f$ is harmonic, the leading term vanishes because $\Delta f=0$. Hence ${\Gamma }_{9}u-h^{2}f=O(h^6)$, or fourth-order error $O(h^4)$ after dividing by $h^2$. More explicitly ${\Delta }^{3}u=0$ gives $u_{xxxxxx}+u_{yyyyyy}=-3(u_{xxxxyy}+u_{xxyyyy})$, so the leading unnormalized term is $h^6(u_{xxxxyy}+u_{xxyyyy})/180$.

The five-point stencil on the forcing satisfiesThus the correction $h^2{\Gamma }_{5}f/12$ cancels exactly the $h^4\Delta f/12$ residual in part (i), leavingThe modified scheme therefore has fourth-order normalized local accuracy, the same improved order as the harmonic-forcing case, now without assuming $\Delta f=0$. It need not have the same leading error constant.