Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 67 / 4

Put $\nu =k/\delta$ and use the discrete Fourier transform on the infinite grid. A spatial Fourier mode $e^{i(j\theta +\ell \varphi )}$ has time coefficient satisfyingIts amplification polynomial of a multilevel finite difference scheme is ${\zeta }^2-2is\zeta -1$, with rootsFor $|s|<1$ they are distinct and have modulus one. Since $|s|\le 2\nu$, a fixed $0<\nu <1/2$ bounds their separation below by $2\sqrt{1-4{\nu }^2}$. Solving for the two mode amplitudes therefore bounds every power of the two-level companion matrix uniformly in frequency and step number. Parseval identity transfers this bound to the discrete L2 norm.

If $\nu >1/2$, an open set of frequencies around $(\pi /2,\pi /2)$ has $|s|>1$ and a root of modulus greater than one. Fourier data supported there grow exponentially. At $\nu =1/2$, the central frequency has the double root $i$, and the recurrence admits $v_n=(A+Bn)i^n$. Thus the root moduli being one do not suffice: a nontrivial Jordan block makes the two-level powers grow linearly. Although this exact frequency has measure zero on the infinite grid, the matrices depend continuously on frequency. For every step count, a neighborhood has powers nearly as large, so their essential supremum is still unbounded. Wave packets in those neighborhoods rule out a uniform stability of a numerical method estimate in the discrete L2 norm.

For arbitrary perturbations in both starting levels the sharp positive range is thereforeThe zero-step limiting recurrence is bounded but does not advance time. A specially filtered startup can suppress a repeated-root mode; that restriction does not establish stability of the full two-level scheme. This is the two-dimensional leapfrog stability threshold.

Divide the recurrence by $2k$ and substitute a smooth exact solution. Centered Taylor expansions in time and space giveThus the leapfrog advection scheme is consistent with normalized residual $O(k^2+{\delta }^2)$. Its unnormalized defect is $O(k^3+k{\delta }^2)$. For a fixed $0<\nu <1/2$, the uniform two-level bound proved above and accumulation over $O(1/k)$ steps yield, for a sufficiently smooth solution,Both starting levels must be supplied consistently; errors of order $k^2+{\delta }^2$ give second-order convergence. For general square-integrable functions as initial data, point sampling is not defined on equivalence classes. Use cell averages or another bounded grid projection, together with a bounded consistent startup. Approximate the initial data by smooth functions, apply the smooth-data convergence estimate, and use the uniform numerical and exact translation bounds to pass to the L2 space limit. This is the content of Lax equivalence theorem in the present setting.

At $\nu >1/2$ the growing Fourier mode prevents general convergence. At the endpoint, initial data supported in increasingly narrow neighborhoods of the double root can have a second-level error of norm $O(k)$ but a final error of order one after $O(1/k)$ steps. These are vanishing starting errors, so the endpoint fails general two-level convergence as well. Consequentlyis also the general convergence range, with a consistent stable initialization. Claims for a particular specially chosen endpoint startup would be a different, restricted assertion.