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

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.