Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 69 / 5

Take the endpoint values $u_0=u_M=0$ and evolve the interior indices $1,\dots ,M-1$. This matches $\Delta x=1/M$ on $[0,1]$. If the printed upper index $m=M$ is retained, the odd ghost extension $u_{M+1}=-u_{M-1}$ makes its boundary equation identically compatible; it does not add an interior unknown.

Writing $h=\Delta x$ and $\tau =\Delta t$, this is the theta method with parameter $a$:For a smooth exact heat solution, its normalized residual at $t_n$ isThus the generic consistency orders are first in time and second in space for $a\ne 1/2$, and second in both time and space for $a=1/2$, the Crank-Nicolson method. These are independent-step orders. Along the special parabolic refinement $\tau =\mu h^2$, the leading combined residual cancels if $a=1/2-1/(12\mu )$, giving a higher combined smooth-data order; it does not change the generic temporal order at independently varied meshes.

The given L2 initial condition need not have pointwise values or four derivatives. Initialize by a bounded cell-average or projection operator, as in L2-compatible initialization of grid data. The Taylor estimate applies to smooth data or sufficiently positive times after heat smoothing, rather than an unqualified uniform high-order estimate at the initial instant for arbitrary rough data. Stability and convergence for rough data are distinct from this consistency order.

The Dirichlet difference Laplacian has an orthogonal sine basis with eigenvalues $-{\eta }_j/h^2$, where ${\eta }_j=4{\sin }^2(j\pi /(2M))$ for $j=1,\dots ,M-1$. On a mode the theta method amplification isIf $a\ge 1/2$, the denominator is positive andThus $|G(s)|\le 1$ for every mode and every $\mu >0$. Orthogonality gives a uniform discrete L2 norm contraction, not merely individual eigenvalue estimates.

If $a<0$, a positive $s=-1/a$ makes the implicit matrix singular. If $a=0$, $G=1-s$ becomes arbitrarily large. If $0<a<1/2$, $G(s)$ tends to $-(1-a)/a$, whose magnitude is greater than one. Hence unconditional stability holds exactly for $a\ge 1/2$.

For completeness, $a<1/2$ is contractive under $\mu {\eta }_{\max }\le 2/(1-2a)$; the mesh-independent sufficient restriction is $\mu \le 1/[2(1-2a)]$. This follows from the same squared-modulus difference and keeps the denominator positive even for negative $a$. At $a=1/2$, very high frequencies approach amplification $-1$, so A-stability does not imply damping of rough heat data. At $a=1$ the Backward Euler method damps them strongly; this is the rough-data damping distinction for the theta method.