 Solution

Write $h=\Delta x$, $k=\Delta t$ and ${\delta }_{xx}{u}_i=(u_{i+1}-2u_i+u_{i-1})/h^2$. Insert a smooth exact heat equation solution into the Crank-Nicolson method. Expanding about the midpoint $(x_i,t_n+k/2)$ givesThus the residual, defined as left side minus right side, isSince $u_t=D{u}_{xx}$ implies $u_{ttt}=D{u}_{xxtt}$,This proves second-order numerical consistency in both space and time. About the old time level, uncombined Taylor remainders can include $O(k^3)$; the midpoint expression displays the symmetric leading errors most cleanly. Multiplying by $k$ gives the one-step defect $O(k^3+k{h}^2)$. Stable evolution with suitable initial and boundary data then gives global error $O(k^2+h^2)$; consistency alone does not prove global convergence.