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.