Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 66 / 5

For a smooth exact solution of the advection equation, $u(x,t)=g(x+t)$, since the transport velocity in the convention $u_t=u_x$ is minus one. Put $d=\Delta x$, $k=\mu d$ and move the scheme's right side to the left. Substitution gives the exact-solution residualIts constant, linear and quadratic Taylor expansion terms cancel. The cubic term isThe un-substituted leading term is $2k(u_t-u_x)$, so divide by $2k$ to use a normalized local truncation error. For a fixed positive Courant ratio,Thus the generic method is second order, provided stability and a compatible second-order starter are supplied.

There are two special ratios rather than an unnoticed higher generic order. At $\mu =1/2$, the scheme is $u_m^{n+1}=u_{m+1}^{n-1}$; both sides lie on the same exact characteristic, and the residual vanishes identically. At $\mu =1$ the previous-level term cancels the unshifted current term for exact data, again producing zero residual on every exact characteristic solution. The first special case is stable, while the second is not stable for arbitrary two-level perturbations, as part (b) shows. Exact propagation of specially initialized data is not a substitute for stability.

Use the Fourier transform for the spatial Cauchy problem, or its periodic analogue, and regard the two starting levels as independently perturbed data. A Fourier mode with spatial factor $e^{im\theta }$ has amplification roots $G$ satisfyingPut $c=2\mu -1$ and $G=e^{i\theta /2}q$. ThenIf $|c|<1$, both roots have modulus one and their separation is bounded below uniformly in frequency:The uniform power bound from separated amplification roots now controls the two-level companion matrix for every time step. Its entries are uniformly bounded, and its eigenvector conditioning is bounded by the reciprocal root gap. The Parseval identity transfers this frequency-uniform bound to the spatial ${\ell }^2$ norm. This proves stability, rather than merely checking each root's modulus.

If $|c|>1$, the frequency $\theta =\pi$ has a root outside the unit disk, so there is exponential instability. If $|c|=1$, the two roots at $\theta =\pi$ coincide on the unit circle. The companion matrix is not a scalar matrix and has a nontrivial Jordan block; its powers grow linearly in the number of steps. Frequencies arbitrarily near that value produce the same lack of a uniform bound for localized Fourier packets, so this also invalidates Cauchy ${\ell }^2$ stability, not only periodic plane-wave stability. At $\mu =1$ the double amplification root is $-1$, and at $\mu =0$ it is $1$.

Therefore the full two-level stability range for a fixed positive Courant ratio isThe endpoint $\mu =0$ is moreover not a positive time step. Bounds deteriorate as $\mu$ approaches either endpoint; the displayed range is not a uniform claim over ratios arbitrarily close to one. A prescribed starter that removes one special parasitic component can change behavior for selected initial data, but does not establish the requested unrestricted two-level stability.