Scalar Equations with Variable Coefficients

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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

.25in
     

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.

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 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 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 condition holds:

Then the spectral approximation (4.3.17) is stable, and the following estimate is fulfilled: