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

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.