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
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
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
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).
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.
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).
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
and stability follows.
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
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 resolution of waves entering through the inflow boundary x = 1. In the constant-coefficient case this resolution requires time steps of size . However, when the inflow boundary is almost characteristic, i.e., when , then the CFL condition is essentially independent of , for (cheb_var.21) boils down to . In purely outflow cases the time step is independent of any resolution requirement at the boundaries, and we are left with the CFL condition restricted solely by the size of the corresponding SL eigenvalues.
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 of the minimal grid size mentioned in point #2 above. We have
. Assume that the following CFL
Then the spectral approximation (4.3.17) is stable, and the following estimate is fulfilled: