Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 66 / 1 / c

Put $k=\Delta t>0$ and retain the same matrix $L$. The Crank--Nicolson method isTake the real mesh-weighted inner product with $U^{n+1}+U^n$. The cross terms cancel, and the identity from part (b) givesThe implicit system is uniquely solvable: if $(I-kL/2)V=0$, thenso $V=0$. Therefore its dissipative Cayley-transform contraction satisfiesIterating proves unconditional stability for every $\mu =k/d^2>0$; the perturbation bound is one and does not depend on the mesh or time-step ratio.

The printed hint's exponential estimate is valid with the logarithmic norm, but its proposed bound by $r(k\alpha [L])$ is not a general inheritance principle. For the trapezoidal stability function $r(z)=(1+z/2)/(1-z/2)$, that real number can even be negative when $k\alpha [L]<-2$, whereas a norm is nonnegative. The direct energy proof above establishes the required result without that assertion.