We consider the inhomogeneous scalar hyperbolic equation

which is augmented with inhomogeneous data prescribed at the inflow boundary

Using forward Euler time-differencing, the spectral
approximation of (cheb_inhomo.1)
reads, at the *N* zeros of ,

and is augmented with the boundary condition

In this section, we study the stability of
(cheb_inhomo.3a), (cheb_inhomo.3b)
in the two cases of

and the closely related

To deal with the inhomogeneity of the boundary condition
(cheb_inhomo.3b), we
consider the
-polynomial

If we set

then satisfies the inhomogeneous equation

which is now augmented by the homogeneous boundary condition

theorem 4.1
together with Duhammel's principle provide us with an a priori
estimate of in terms of the initial and the
inhomogeneous data, and
.
Namely, if the CFL condition (meth_cheb.12)
holds, then we have

Since the discrete norm is supported at the
zeros of , where , we conclude

The last theorem provides us with an a priori
stability estimate in terms of the
initial data, , the inhomogeneous data, , and
the boundary data *g*(*t*). The dependence on the boundary data involves the
factor of , which grows
linearly with *N*, so that we end up with
the stability estimate

An inequality similar to (cheb_inhomo.12)
is encountered in the stability study of
finite difference approximations to mixed initial-boundary hyperbolic systems.
We note in passing that the stability estimate
(cheb_inhomo.12) together with the
usual consistency requirement guarantee the spectrally accurate convergence of
the spectral approximation.

Thu Jan 22 19:07:34 PST 1998