 Solution

The Gershgorin circle theorem says that every eigenvalue of a square matrix lies in at least one disc centred at a diagonal entry with radius the sum of the absolute values of the off-diagonal entries in that row.

For interior unknowns write the update as $u^{n+1}=B{u}^n$. The matrix is real symmetric and tridiagonal, with diagonal $1-\mu (a_{m-1/2}+a_{m+1/2})$ and adjacent entries $\mu a_{m+1/2}$. Each Gershgorin radius is at most $\mu (a_{m-1/2}+a_{m+1/2})$. Its real interval is therefore contained inFor $0<\mu \le 1/(2a_{+})$, every eigenvalue lies in $[-1,1]$. Symmetry gives an orthogonal diagonalization, so $\Vert B^n{\Vert }_2\le 1$ for every $n$: errors in initial data cannot grow. This is stability in the discrete ${\ell }^2$ norm, also in its $h$-weighted version. One can additionally see maximum-norm stability directly: the update weights are nonnegative with row sums at most one once the zero boundary unknowns are removed. Hence $\Vert B{\Vert }_{\infty }\le 1$.