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Epilogue

On previous subsections we analyzed the stability of Fourier method in terms of two main ingredients: weighted -stability on the one hand, and high frequencies instability on the other hand. Here we would like to show how both of these ingredients contribute to the actual performance of the Fourier method.

We first address the issue of resolution. We were left with the impression that the weak -instability is a rather 'rare occurrence', as it is excited only in the presence of nonsmooth initial data. But in fact, the mechanism of this weak -instability will be excited whenever the Fourier method lacks enough resolution.
In this context let us first note that the solution of the underlying hyperbolic problem may develop large spatial gradients due to the almost impinging characteristics along the zeroes of the increasing part of a(x). Consequently, the Fourier method might not have enough modes to resolve these large gradients as they grow in time. This tells us that independent whether the initial data are smooth or not, the computed approximation will then 'see' the underlying solution as a nonsmooth one, and this lack of resolution will be recorded by a slower decay of the computed Fourier modes. The latter will experience the high-frequency instability discussed earlier and this in turn will lead to the linear -growth. Our prototype example of is case in point: according to Corollary 3.2, one needs here at least modes in order to resolve the solution, for otherwise, (weak-in.19) shows that spurious oscillations will contaminate the whole computed spectrum.

Figure: Fourier solution of .

Figure: Fourier solution of .

Figure: Fourier solution of .

Figure: Fourier solution of .

Figure: Fourier solution of .

We conclude that the lack of resolution manifests itself as a weak -instability. This phenomenon is demonstrated in Figures 3.5-3.9, describing the Fourier method (weighted.6) subject to (the perfectly smooth ...) initial condition, . Figure 3.5 shows how the Fourier method with fixed number of N=64 modes propagates information regarding the steepening of the Fourier solution in physical space, from low modes to the high ones. And, as this information is being transferred to the high modes, their amplification become more noticeable as time progresses in Figures 3.5a-3.5d. Consequently, though N=64 modes are sufficient to resolve the exact solution at , Figure 3.6c-d shows that at later time, t=3 and in particular t=5, the under resolved Fourier solution with 64-modes will be completely dominated by the spurious centered spike. This loss of resolution requires more modes as time progresses. Figure 3.7 shows how the Fourier method is able to resolve the exact solution at t=3.5, once 'sufficiently many' modes, are used, in agreement with Corollary 3.3. According to Figures 3.8 and 3.9, modes are required to correctly resolve the two strong boundary dipoles at t=4, yet at t=8 the Fourier solution will be completely dominated by the spurious centered spike.

Assuming that the Fourier method contains sufficiently many modes dictated by the requirement of resolution, we now turn to the second issue of this section concerning the convergence of the Fourier method.

. The requirement from the initial data to have at least -regularity is clearly necessary in order to make sense of its pointwise interpolant.    Next: Skew-Symmetric Differencing Up: AliasingResolution and (weak) Previous: Algebraic stability and weak