Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 40D (Numerical Analysis)

Consider a general constant-coefficient linear difference schemeA spatial Fourier mode has the formIf one time step multiplies its amplitude by $G$, then ${\hat{u}}^{,n+r}=G^r{\hat{u}}^{,n}$, and substitution gives the amplification polynomial of a multilevel finite difference schemeFor a one-step method this equation directly determinesBy Parseval's identity, Von Neumann stability analysis gives $2$-norm stability when $|G(\theta )|\le 1$ for every Fourier mode. For a multilevel method, every root must obey the corresponding root condition.

Insertinto the diffusion scheme. Sincewe obtainThus the Crank-Nicolson diffusion scheme hasFor $\mu \ge 0$, the denominator is positive andHence

For the wave scheme, use $v_m^n=G^n{e}^{im\theta }$. After multiplying by $G$, the recurrence becomesEquivalently,Because the two roots have product one, both lie on the unit circle precisely whenThis must hold for every $s\in [0,1]$. At $s=0$ it requires $\rho \le 1$. Since the question assumes $1\le \rho \le 2$, stability is possible only for $\rho =1$. The condition at $s=1$ then becomesor $0\le \mu \le 1$. With the stipulated $\mu >0$, the centered three-level wave scheme is stable exactly whenFor every $\rho >1$, sufficiently long-wavelength modes have a real root larger than one, whatever the value of $\mu$.

Let $L$ be the tridiagonal matrix with diagonal entries $-2$ and adjacent off-diagonal entries $1$ on the interior grid points. The finite-interval scheme isFor $J$ interior points, the dirichlet discrete Laplacian has eigenvaluesThe corresponding eigenvalues of the amplification matrix areSince ${\lambda }_j<0$ and $\mu \ge 0$,for every $j$. The matrix is real symmetric in the sine eigenbasis, so its induced $2$-norm is the largest $|q_j|$. This proves the crank-Nicolson stability on a finite Dirichlet interval and gives