If a simple symmetric hyperbolic system with constant coefficients in one dimension of the form
with symmetric, is discretized in space by approximating the space derivative with an operator
satisfying SBP, a semi-discrete energy estimate can be derived, modulo boundary conditions (discussed in
Section 10). For this, we define
Taking a time derivative, using the symmetry of , the fact that and commute because the
latter has constant coefficients, and the SBP property,
Therefore, modulo the boundary terms, an energy estimate and semi-discrete stability follow. As usual, the
addition of lower-order undifferentiated terms to the right-hand side of Eq. (8.36) still gives an energy
estimate at the semi-discrete level, modulo boundary terms.
When considering variable coefficients, Eq. (8.38) becomes
and the commutator between and needs to be uniformly bounded for all resolutions in order to
obtain an energy estimate.
Estimates for the term involving the commutator have been given in [317, 319, 318, 407]. In
order to discuss them, we first notice that the SBP property of with respect to and the
symmetry of imply that the operator is symmetric with respect to ,
for all grid functions and . Therefore, its norm is equal to its spectral radius and we have the
for all grid functions . Hence, the problem is reduced to finding an upper bound for the spectral radius
of , which is independent of the scalar product .
Next, in order to find such an upper bound, we write the FD operator as
where the ’s are the coefficient of a banded matrix; that is, there exists , which is independent
of such that for . Then, we have
from which it follows, under the assumption that is continuously differentiable and that its derivative is
bounded, , , where .
Now we can easily estimate the spectral radius of , based on the simple observation that
for any norm on the space of grid functions . Choosing the 1-norm we find
from which it follows that
The important point to notice here is that for each fixed , the sum in the expression for involves at
most non-vanishing terms, since for . For the SBP operators and scalar
products used in practice, namely those for which the latter has the structure given by Eq. (8.29),
can be bounded by a constant, which is independent of resolution. Since the spectral
radius of is equal to that of its transposed, we may interchange the roles of and
in the definition of the constant , and with these observations we arrive at the following
- A key ingredient used above to uniformly bound the norm of is that the SBP scalar
products used in practice have the form (8.29). In those cases, both the boundary width and the
boundary stencil size (defined below Example 52) associated with the corresponding difference
operators are independent of . Therefore, the constants and can also be bounded
independently of .
- For the operator defined in Example 51, for instance, Eq. (8.48) gives , ,
and we obtain the optimal estimate corresponding to the one in the continuum limit,
- For the operator defined in Example 52, in turn, Eq. (8.48) gives
- For spectral methods the constants and typically grow with as the coefficients
do not form a banded matrix anymore. This leads to difficulties when estimating the commutator;
see  for a discussion on this point.
- It is also possible to avoid the estimate on the commutator between and altogether
through skew-symmetric differencing , in which the problem is discretized according to
A straightforward energy estimate shows that this leads to strict stability, after the imposition of
appropriate boundary conditions.
- SBP by itself is not enough to obtain an energy estimate since the boundary conditions still need to
be imposed, and in a way such that the boundary terms in the estimate after SBP are under control.
This is the topic of Section 10.