Consider a discrete grid consisting of points and uniform spacing
on some, possibly
unbounded, domain
.
Definition 17. A difference operator approximating
is said to satisfy SBP on the domain
with respect to a positive definite scalar product
,
This is the discrete counterpart of integration by parts for the operator,
If the interval is infinite, say or
, certain fall-off conditions are required and
Eq. (8.22
) replaced by dropping the corresponding boundary term(s).
Example 50. Standard centered differences as defined by Eq. (8.19) in the domain
or for
periodic domains and functions satisfy SBP with respect to the trivial scalar product (
),
The scalar product or associated norm are said to be diagonal if
that is, if
When constructing SBP operators, the discrete scalar product cannot be arbitrarily fixed and afterward
the difference operator solved for so that it satisfies the SBP property (8.22) – in general this leads to no
solutions. The coefficients of
and those of
have to be simultaneously solved for. The resulting
systems of equations lead to SBP operators being in general not unique, with increasing freedom with the
accuracy order. In the diagonal case the resulting norm is automatically positive definite but not so in the
full-restricted case.
We label the operators by their order of accuracy in the interior and near boundary points. For diagonal
norms and restricted full ones this would be and
, respectively.
Example 51. : For the simplest case,
, the SBP operator and scalar product are unique:
The operator and its associated scalar product are also unique in the diagonal norm case:
Example 52. :
On the other hand, the operators have one, three and ten free parameters,
respectively. Up to
their associated scalar products are unique, while for
one of the free
parameters enters in
. For the full-restricted case,
have three, four and five free
parameters, respectively, all of which appear in the corresponding scalar products.
A possibility [396] is to use the non-uniqueness of SBP operators to minimize the boundary stencil size
. If the difference operator in the interior is a standard centered difference with accuracy-order
then there are
points at and near each boundary, where the accuracy is of order
(with
in the diagonal case and
in the full restricted one). The integer
can be referred to as the boundary width. The boundary stencil size
is the number of
gridpoints that the difference operator uses to evaluate its approximation at those
boundary
points.
However, minimizing such size, as well as any naive or arbitrary choice of the free parameters, easily
leads to a large spectral radius and as a consequence restrictive CFL (see Section 7) limit in the case of
explicit evolutions. Sometimes it also leads to rather large boundary truncation errors. Thus, an alternative
is to numerically compute the spectral radius for these multi-parameter families of SBP operators and find
in each case the parameter choice that leads to a minimum [399, 281]. It turns out that in this way the
order of accuracy can be increased from the very low one of
to higher-order ones such as
or
with a very small change in the CFL limit. It involves some work, but since the SBP
property (8.22
) is independent of the system of equations one wants to solve, it only needs to be done once.
In the full-restricted case, when marching through parameter space and minimizing the spectral radius, this
minimization has to be constrained with the condition that the resulting norm is actually positive
definite.
The non-uniqueness of high-order SBP operators can be further used to minimize a combination of the average of the boundary truncation error (ABTE), defined below, without a significant increase in the spectral radius. For definiteness consider a left boundary. If a Taylor expansion of the FD operator is written as
then Table 6 illustrates the results of this optimization procedure for theThe coefficients for the SBP operators
and, in particular, for their optimized versions, are available, along with their associated dissipation operators described below in Section 8.5, from the arXiv in [141Operator | Min. bandwidth | Min. ABTE and spectral radius |
Spectral radius | 995.9 | 2.240 |
ABTE | 20.534 | 0.7661 |
Remarks:
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
![]() This work is licensed under a Creative Commons License. E-mail us: |