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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 -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.