There are two main approaches to enforce stability at this point: skew-symmetric differencing and smoothing. We discuss these issues in the next two subsections.
The essential argument of well-posedness for symmetric hyperbolic systems with
constant coefficients is the fact that (say in the 1-D case) is a skew-adjoint operator.
With variable coefficients this is also
true, modulo low-order bounded terms, i.e.,
The stability proofs of spectral methods follow the same line, i.e., we have
in the Fourier space, compare (meth_spec.45),
and stability amounts to show that the second term in (skew.2) is bounded:
for then we have in (skew.2) (as in ((skew.1) )
a skew-adjoint term with an
additional bounded operator. The difficulty with the stability of
pseudo-spectral methods arises from the fact that the second term on the right
of (skew.2) is unbounded,
To overcome this difficulty, we can discretized the symmetric hyperbolic system
(again, say the 1-D case)
when the spatial operator is already put in the ``right''
skew-adjoint form, compare (skew.1),
The pseudospectral approximation takes the form
In the Fourier space, this gives us
Now, is symmetric because
is, is bounded and
stability follows.