Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 40E (Numerical Analysis)

For the discrete Fourier mode $u_m^n=G^n{e}^{im\theta }$, the centred second difference has multiplier $-4{\sin }^2(\theta /2)$. Hence the amplification factor isFor every $\mu >0$, $|G(\theta )|\le 1$. The discrete Parseval identity then shows directly that the discrete $L^2$ norm cannot increase at any time step, proving unconditional stability for the Cauchy problem.

Let $A_h$ be the symmetric Dirichlet second-difference matrix. Its eigenvalues are nonpositive, so every eigenvalue ofhas modulus at most one. Thus $\Vert R_h{\Vert }_{2,h}\le 1$.

Insert the exact smooth solution into the scheme. The local truncation error per step has discrete norm $O(k^3+k{h}^2)$, from the order-three trapezoidal time residual and the order-two centred spatial residual. If $e^n$ is the grid error, stability and iteration giveThere are at most $1/k$ steps and $e^0=0$, henceThis proves convergence for every $\mu =k/h^2>0$ without invoking the Lax equivalence theorem.