Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 341 / 4

The original PDF has a spatial shift $u_{m+2}^{n+2}$, so the printed scheme is not the usual second-order BDF discretization. The PDF does have $-2u_m^{n+2}$ on the right; the TeX aid loses that factor two, and that transcription must not be used.

Let $h=\Delta x$, $k=\Delta t$, and expand an exact smooth heat equation solution about $(x_m,t_{n+2})$. Relative to the standard unshifted backward differentiation formula, the printed newest value addsDividing the raw residual by $2k/3$, its expansion isIn particular, for the usual refinement $k=\mu h^2$ with fixed $\mu >0$, the term $3u_x/(\mu h)$ generally diverges. A concrete smooth counterexample is $u=e^{-{\pi }^{2}t/4}\cos (\pi x/2)$, which satisfies the Dirichlet boundary conditions and has $u_x\ne 0$ at $x=1/2$. Thus the printed scheme has no consistent order under fixed-$\mu$ refinement. Its normalized shift error is $O(h/k)$, so one could only investigate a special coupled limit such as $h=o(k)$ after specifying a valid boundary closure; it has no standard joint second-order accuracy.

For the likely intended correction $u_m^{n+2}$, the extra shift terms disappear. The normalized local truncation error isWith a stable, suitably accurate starter, the corrected backward differentiation formula combined with the second-order central difference therefore hasThis is a correction to the printed method, not a claim about it.

The spatial shift in the PDF also requires a boundary value beyond the supplied right endpoint: if $m$ is the last interior index, the newest $m+2$ value is outside the interval. Thus the printed Dirichlet method has no specified boundary closure, and no unique stability range can be assigned to it as stated.

For completeness, the literal interior formula has an exact von Neumann stability analysis on an infinite or periodic grid. Substitution of a Fourier mode $U_m^n={\xi }^n{e}^{im\theta }$ givesFor the root of a polynomial tending to one as $\theta \to 0$, expansion givesConsequently $\mu <3$ is unstable: sufficiently small nonzero frequencies have a root of a polynomial of modulus greater than one, and arbitrarily fine periodic grids contain such frequencies.

Conversely, if $\mu \ge 3$ and $s=\cos \theta$, thenFor $A=u+iv$ with $u\ge 3$,Since $|A|\ge 3$, the complex quadratic Schur criterion yields both roots of a polynomial in the unit disk, strictly inside except at $\theta =0$, where they are $1$ and $1/3$. Uniform stability follows directly from the BDF2 discrete energy identity: the modal recurrence is $3v^{n+2}-4v^{n+1}+v^n=-(A-3)v^{n+2}$, whose energy decreases because $\mathrm{Re}(A-3)\ge 0$. Summing the modal energies proves a bound independent of the grid and even of $\mu \ge 3$; no eigenvector-separation assumption is needed. This is the stability of a spatially shifted BDF2 stencil. Thus the literal periodic-grid result isThis cannot establish stability of an unspecified Dirichlet boundary condition closure. It also does not restore consistency of a numerical method.

For the intended unshifted backward differentiation formula, a spatial eigenvalue $-\beta \le 0$ givesThe Schur stability criterion gives the root condition for a multistep method for every $k\beta \ge 0$, with strict inequalities for $\beta >0$. A useful uniform energy method proof, including the repeated interior roots that a bare separation argument would miss, is the BDF2 discrete energy identity. With ${\delta }^{2}U=U^{n+2}-2U^{n+1}+U^n$ andthe corrected Dirichlet discrete Laplacian scheme satisfiesThis follows by taking the inner product of $3U^{n+2}-4U^{n+1}+U^n=2k{L}_h{U}^{n+2}$ with $U^{n+2}$ and using summation by parts. It controls both time levels uniformly, regardless of the mesh ratio. Hence the corrected method is stable for