= Solution
Write $h=\Delta x$, impose $u_0=u_{M+1}=0$, and use the discrete <inner product> $\langle u,v\rangle_h=h\sum_{m=1}^M\overline u_m v_m$. Discrete <integration by parts> for the second difference gives
$$
\operatorname{Re}\left[h\sum_{m=1}^M\overline u_m
\frac{u_{m-1}-2u_m+u_{m+1}}{h^2}\right]
=-\frac1h\sum_{m=0}^M|u_{m+1}-u_m|^2.
$$
For the centered drift, reindexing with the zero endpoints gives
$$
\operatorname{Re}\sum_{m=1}^M\overline u_m(u_{m+1}-u_{m-1})=0,
$$
since the two sums are complex conjugates. This is the fact that <centered discrete advection is skew-adjoint>. The semidiscrete <energy estimate> is therefore
$$
\boxed{\frac12\frac d{dt}\|u\|_h^2
=-\frac1h\sum_{m=0}^M|u_{m+1}-u_m|^2\leq0.}
$$
The finite-dimensional linear system has a unique solution for all times, and \b[$\|u(t)\|_h\leq\|u(0)\|_h$], uniformly in the mesh and in real $\kappa$. Differences of two semidiscrete solutions obey the same bound, which is the required stability.
One may also use the smallest <eigenvalue> $\lambda_h=4h^{-2}\sin^2(\pi/[2(M+1)])$ of the negative second-difference operator to obtain $\|u(t)\|_h\leq e^{-\lambda_ht}\|u(0)\|_h$. This conclusion concerns the <method of lines>; a later choice of time integrator has its own stability conditions.
Back to article page
