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The Spectral Fourier Approximation

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We begin with the simplest hyperbolic equation - the scalar constant-coefficients wave equation

subject to initial conditions

and periodic boundary conditions.

This Cauchy problem can be solved by the Fourier method: with we obtain after integration of (meth_spec.1),

with solution

and hence

Thus the solution operator in this case amounts to a simple translation

This is reflected in the Fourier space, see (meth_spec.4), where each of the Fourier coefficients has the same change in phase and no change in amplitude; in particular, therefore, we have the a priori energy bound (conservation)

We want to solve this equation by the spectral Fourier method. To this end we shall approximate the spectral Fourier projection of the exact solution . Projecting the equation (meth_spec.1) into the N-space we have

Since commutes with multiplication by a constant and with differentiation we can write this as

Thus satisfies the same equation as the exact solution does, subject to the approximate initial data

The resulting equations amount to 2N + 1 ordinary differential equations (ODEs) for the amplitudes of the projected solution

subject to the initial conditions

Since these equations are independent of each other, we can solve them directly, obtaining

and the approximate solution takes the form

Hence, the approximate solution satisfies

and therefore, it converges spectrally to the exact solution, compare (app_fourier.26),

Similar estimates holds for higher Sobolev norms; in fact if the initial data is analytic then the convergence rate is exponential. In this case the only source of error comes from the initial data, that is we have the error equation

subject to initial error

Consequently, we have the a priori estimate of this constant coefficient wave equation

Now let us turn to the scalar equation with variable coefficients

This hyperbolic equation is well-posed: by the energy method we have

and hence

with

In other words, we have for the solution operator

and similarly for higher norms. As before, we want to solve this equation by the spectral Fourier method. We consider the spectral Fourier projection of the exact solution ; projecting the equation (meth_spec.19) we get

Unlike the previous constant coefficients case, now does not commute with multiplication by a(x,t), that is, for arbitrary smooth function we have (suppressing time dependence)

while

Thus, if we exchange the order of operations we arrive at

While the second term on the right is not zero, this commutator between multiplication and Fourier projection is spectrally small, i.e.,

and so we intend to neglect this spectrally small contribution and to set as an approximate model equation for the Fourier projection of u(x,t)

The second term may lie outside the N-space, and so we need to project it back, thus arriving at our final form for the spectral Fourier approximation of (meth_spec.19)

Again, we commit here a spectrally small deviation from the previous model, for

The Fourier projection of the exact solution does not satisfy (meth_spec.22a)-(meth_spec.22b), but rather a near-by equation,

where the local truncation error, is given by

The is the amount by which the (projection of) the exact solution misses our approximate mode (meth_spec.27); in this case it is spectrally small by the errors committed in (meth_spec.26) and (meth_spec.18). More precisely we have

depending on the degree of smoothness of the exact solution. We note that by hyperbolicity, the later is exactly the degree of smoothness of the initial data, i.e., by the hyperbolic differential energy estimate

and in the particular case of analytic initial data, the truncation error is exponentially small.

From this point of view, the spectral approximation (meth_spec.27) satisfies an evolution model which deviates by a spectrally small amount from the equation satisfied by the Fourier projection of the exact solution (meth_spec.29). This is in addition to the spectrally small error we commit initially, as we had before    Next: Stability and convergence Up: THE FOURIER METHOD Previous: THE FOURIER METHOD