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Problems with inhomogeneous initial-boundary conditions

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 tex2html_wrap_inline11151-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.

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998