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When dealing with finite difference approximations
which are *locally supported*, i.e.,
finite difference schemes whose stencil occupy a *finite* number of
neighboring grid cells each of which of size , then one
encounters the hyperbolic CFL stability restriction

With this in mind, it is tempting to provide a heuristic justification for the
stability of spectral methods, by arguing that a CFL stability restriction
similar to (cheb_var.1) should hold. Namely, when
is replaced by the minimal
grid size, , then (cheb_var.1) leads
to

Although the final conclusion is correct (consult (meth_cheb.16)),
it is important to
realize that this ``handwaving'' argument is not well-founded in the case of
spectral methods. Indeed, since the spectral stencils occupy the whole
interval (-1,1), spectral methods do not lend themselves to the stability
analysis of locally supported finite difference approximations. Of course,
by the same token, this explains the existence of *unconditionally*
stable fully implicit (and hence globally supported) finite difference
approximations.

As noted earlier, our stability proof (in Theorem (4.1)) shows that the CFL condition (cheb_var.2) is related to the following two points:

.25in #1. The size of the corresponding Sturm-Liouville eigenvalues, . .25in #2. The minimal gridsize, .

The second point seems to support the fact that plays an essential role in the CFL stability restriction for the global spectral methods, as predicted by the local heuristic argument outlined above. To clarify this issue we study in this section the stability of spectral approximations to scalar hyperbolic equations with variable coefficients. The principal raison d'tre, which motivates our present study, is to show that our stability analysis in the constant coefficients case is versatile enough to deal with certain variable-coefficient problems.

We now turn to discuss scalar hyperbolic equations with positive variable
coefficients,

which are augmented with homogeneous conditions at the inflow boundary

We consider the dospectral Jacobi method collocated at the *N*
zeros of . Using forward Euler time-
differencing, the resulting approximation reads

together with the boundary condition

Arguing along the lines of Theorem (4.1), we have

**PROOF**. We divide (cheb_var.13a) by ,

and, proceeding as before, we square both sides to obtain

The first expression, I, involves discrete summation of the
-polynomial and
since (in view of (cheb_var.13b)),
the *N*-nodes Gauss-Lobatto
quadrature rule yields

We integrate by parts the right-hand side of I, substitute with and a straightforward integration by parts yields

The second expression, II, gives us

The inverse inequality (4.1.31) with weight
implies

and the expression does not exceed

Consequently, we have

Equipped with (cheb_var.17) and (6.19) we return to (6.16) to find

and (cheb_var.15b) follows in view of the CFL condition (6.14b).

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1. The case corresponds to one
variant of the stability statement of theorem 4.1.
Similar stability statements with the
appropriate weights which correspond to various alternatives of theorem
4.1, namely, with
, and
, hold. These statements
cover the stability of the corresponding spectral and dospectral Jacobi
approximations with variable coefficients.
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2. We should highlight the fact that the
stability assertion stated in theorem 4.3
depends solely on the uniform bound of but otherwise is
independent of the smoothness of a(x).
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3. The proof of theorem 4.3
applies mutatis mutandis to the case of variable
coefficients with a = a(x,t). If are -functions in
the time variable, then (cheb_var.20) is replaced by
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and stability follows.

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4. We conclude by noting that the CFL condition
(cheb_var.14b) depends on the
quantity , rather than the
minimal grid size, , as in the constant-coefficient
case
(compare (meth_cheb.12)).
This amplifies our introductory remarks at the beginning of
this section, which claim that the stability restriction is
essentially due to the size of the Sturm-Liouville eigenvalues, . Indeed, the other portion of the CFL condition, requiring
guarantees the __

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We close this section with
the particular example
Observe that no augmenting
boundary conditions are required,
since both boundaries, , are outflow ones.
Consequently, the various forward Euler -spectral approximations
in this case amount to
The CFL stability restriction in this case is related to
the -size of the Sturm-Liouville eigenvalues (point #1 above),
but otherwise it is __

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. Assume that the following CFL
condition
holds:
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Then the spectral approximation (4.3.17) is stable, and the following estimate is fulfilled:

Thu Jan 22 19:07:34 PST 1998