Consider a discrete grid consisting of points and uniform spacing on some, possibly unbounded, domain .
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 ifrestricted full if independent of .
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  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 for more details.
The coefficients for the SBP operators and also as complete source code from the Einstein Toolkit .
|Operator||Min. bandwidth||Min. ABTE and spectral radius|
Living Rev. Relativity 15, (2012), 9
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