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Discrete Lie derivative. (English) Zbl 1387.58004

Karasözen, Bülent (ed.) et al., Numerical mathematics and advanced applications – ENUMATH 2015. Selected papers based on the presentations at the European conference, Ankara, Turkey, September 14–18, 2015. Cham: Springer (ISBN 978-3-319-39927-0/hbk; 978-3-319-39929-4/ebook). Lecture Notes in Computational Science and Engineering 112, 635-643 (2016).
Summary: Convection is an important transport mechanism in physics. Especially, in fluid dynamics at high Reynolds numbers this term dominates. Modern mimetic discretization methods consider physical variables as differential \(k\)-forms and their discrete analogues as \(k\)-cochains. Convection, in this parlance, is represented by the Lie derivative, \(\mathcal{L}_{X}\). In this paper we design reduction operators, \(\mathcal{R}\) from differential forms to cochains and define a discrete Lie derivative, \(\mathsf L_{X}\) which acts on cochains such that the commutation relation \(\mathcal{R}\mathcal{L}_{X} = \mathsf{L}_{X}\mathcal{R}\) holds.
For the entire collection see [Zbl 1358.65003].

MSC:

58A10 Differential forms in global analysis
65D25 Numerical differentiation
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