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Therefore the leading terms of truncation error can be eliminated, we can get the following theorem.
By Theorem 1, we may assume that the leading terms of the elements of S are distinct.
Since such deviations do not affect the leading terms of the linear stability analysis (as it can be verified a posteriori) they will be simply neglected.
The key ingredient of the theory is the so-called composition lemma which characterizes the leading terms of elements in the given ideal.
Similarly, compared with the truncation error (T_{6}) of scheme (6) and the truncation error (T_{7}) of scheme (7), the signs of leading terms of (T_{6}) and (T_{7}) are opposite.
Compared with the truncation error (T_{5}) of scheme (5) and the truncation error (T_{8}) of scheme (8), the signs of leading terms of (T_{5}) and (T_{8}) are opposite.
Similar(50)
Note that the HO leading term of the capacity expansion from Eq. (15b) also results in ; however, the subsequent (all positive) items in the series wipe out the extremum.
Both leading terms are independent of the strength of anisotropy.
It is also interesting to note that both leading terms are independent of the size of anisotropy.
Our Nyström method has made use of the cancellation of leading terms in the series expansion of integral kernels to handle hyper singularities in near terms.
These can be leading terms in power expansions of small parameters.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com