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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