The diffusion equation (16.1) is solved by the fully discretized scheme
\(u_{\ell}^{n+1}-\frac{1}{2}(\mu-\zeta)\left(u_{\ell-1}^{n+1}-2
u_{\ell}^{n+1}+u_{\ell+1}^{n+1}\right)=u_{\ell}^{n}+\frac{1}{2}(\mu+\zeta)\left(u_{\ell-1}^{n}-2
u_{\ell}^{n}+u_{\ell+1}^{n}\right)\)
(16.61)
where \(\zeta\) is a given constant. Prove that (16.61) is a second-order method
for all \(\zeta \neq \frac{1}{6}\), while for the choice \(\zeta=\frac{1}{6}\)
(the Crandall method) it is of order 4 .
