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Difference forms

Mansfield, Elizabeth L., Hydon, Peter E. (2008) Difference forms. Foundations of Computational Mathematics, 8 (4). pp. 427-467. ISSN 1615-3375. (doi:10.1007/s10208-007-9015-8) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:15520)

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Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of "discrete differential forms" built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes.

Item Type: Article
DOI/Identification number: 10.1007/s10208-007-9015-8
Uncontrolled keywords: difference forms; lattice variety; cohomology; difference chains; local exactness; local difference potentials
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Elizabeth Mansfield
Date Deposited: 23 Sep 2009 10:13 UTC
Last Modified: 28 May 2019 13:53 UTC
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