Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 69 / 2

Write $h=\Delta x$, $k=\Delta t$, and assume $a,u$ are sufficiently smooth for the following Taylor expansions. The symmetric half-grid diffusion consistency calculation giveswhereThe normalized local truncation error of the full update is consequentlyThe PDF prints an advection equation, $u_t=a(x)u_x$, whereas this stencil approximates conservative diffusion, $u_t=(a{u}_x{)}_x$. For the literal printed equation the leading defect is $(a-a^{\prime })u_x-a{u}_{xx}$, which generally does not vanish: the method is inconsistent, with an $O(1)$ normalized defect.

For the intended variable-coefficient conservative diffusion equation, the first two terms cancel and ${\mathcal{T}}_{h,k}=O(k+h^2)$. Under the parabolic step scaling $k=O(h^2)$, this is $O(h^2)$, while the unnormalized one-step local truncation error is $k{\mathcal{T}}_{h,k}=O(h^4)$. These two orders use different normalizations. With numerical stability and compatible smooth initial and boundary conditions, the intended diffusion problem has global error $O(k+h^2)$ on a fixed time interval. Positivity and boundedness of $a$ alone do not justify the displayed Taylor expansion; coefficient regularity is also needed for that error formula.

Put $r=k/h^2$. The update isUnder $k\le h^2/(2\beta )$, all three weights are nonnegative and sum to one. With the imposed zero Dirichlet boundary conditions, this is a convex combination including any boundary values. Thus the monotone half-grid diffusion update givesDifferences of two computed solutions obey the same estimate, proving numerical stability in the discrete maximum norm, uniformly in $h,k,n$.

There is also an L2 norm proof. The diffusion matrix $L_h$ is symmetric and satisfiesSince $(v_{m+1}-v_m{)}^2\le 2(v_{m+1}^2+v_m^2)$, every eigenvalue lies in $[-4\beta /h^2,0]$. Hence the eigenvalues of $I+k{L}_h$ lie in $[-1,1]$ under the same step restriction, and the spectral theorem for real symmetric matrices gives an L2 norm contraction. These stability estimates are valid for the printed recurrence, although they cannot repair its inconsistency with the printed advection equation.