Abstract
We present a discretization of the linear advection of differential forms on bounded domains. The framework established in [4] is extended to incorporate the Lie derivative, \(\mathcal{L}\), by means of Cartan’s homotopy formula. The method is based on a physics-compatible discretization with spectral accuracy. It will be shown that the derived scheme has spectral convergence with local mass conservation. Artificial dispersion depends on the order of time integration.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R. Abraham, J. E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis, and Applications, volume 75 of Applied Mathematical Sciences. Springer, 2001.
M. I. Gerritsma, R. Hiemstra, J. J. Kreeft, A. Palha, P. Rebelo, and D. Toshniwal. The geometric basis of numerical methods. Proceedings of ICOSAHOM 2012 (this issue).
Marc Gerritsma. Edge functions for spectral element methods. In Jan S. Hesthaven and Einar M. Rønquist, editors, Spectral and High Order Methods for Partial Differential Equations, volume 76 of Lecture Notes in Computational Science and Engineering, pages 199–207. Springer Berlin Heidelberg, 2011.
J. Kreeft, A. Palha, and M. I. Gerritsma. Mimetic framework on curvilinear quadrilaterals of arbitrary order. Arxiv preprint arXiv:1111.4304, 2011.
A. Palha. High order mimetic discretization: Development and application to Laplace and convection-diffusion problems in arbitrary quadrilaterals. PhD thesis, TUDelft, 2013.
A. Palha, P. Pinto Rebelo, R. Hiemstra, J. Kreeft, and M. I. Gerritsma. Physics-compatible discretization techniques on single and dual grids, with application to the poisson equation of volume forms. Submitted to J. Comp Phys., 2012.
N. Robidoux. Polynomial Histopolation, Superconvergent Degrees Of Freedom, And Pseudospectral Discrete Hodge Operators. Unpublished: http://www.cs.laurentian.ca/nrobidoux/prints/super/histogram.pdf, 2008.
Morita Shigeyuki. Geometry of differential forms, volume 201 of Translations of mathematical monographs. American mathematical society, 2001.
Acknowledgements
The authors would like to thank the valuable comments of both reviewers and the funding received by FCT – Foundation for science and technology Portugal through SRF/BD/36093/2007 and SFRH/BD/79866/2011.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Palha, A., Rebelo, P.P., Gerritsma, M. (2014). Mimetic Spectral Element Advection. In: Azaïez, M., El Fekih, H., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012. Lecture Notes in Computational Science and Engineering, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-01601-6_26
Download citation
DOI: https://doi.org/10.1007/978-3-319-01601-6_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01600-9
Online ISBN: 978-3-319-01601-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)