Abstract
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, L X which acts on cochains such that the commutation relation \(\mathcal{R}\mathcal{L}_{X} = \mathsf{L}_{X}\mathcal{R}\) holds.
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Notes
- 1.
The vector space \(T_{p}(\mathcal{M})\) and \(T_{p}^{{\ast}}(\mathcal{M})\) are ismorphic, but there is no natural isomorphism. One way to associate vectors v to covectors α is by means of the metric tensor: α i  = g ij v j. With this association we have v â™â€‰= α and α ♯ = v. By construction the musical operators are metric-dependent.
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Gerritsma, M., Kunnen, J., de Heij, B. (2016). Discrete Lie Derivative. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_61
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DOI: https://doi.org/10.1007/978-3-319-39929-4_61
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