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Improved approximation of phase-space densities on triangulated domains using discrete flow mapping with p-refinement

Bajars, Janis; Chappell, David; Hartmann, Timo; Tanner, Gregor

Authors

Janis Bajars

David Chappell

Timo Hartmann

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GREGOR TANNER GREGOR.TANNER@NOTTINGHAM.AC.UK
Professor of Applied Mathematics



Abstract

We consider the approximation of the phase-space flow of a dynamical system on a triangulated surface using an approach known as Discrete Flow Mapping. Such flows are of interest throughout statistical mechanics, but the focus here is on flows arising from ray tracing approximations of linear wave equations. An orthogonal polynomial basis approximation of the phase-space density is applied in both the position and direction coordinates, in contrast with previous studies where piecewise constant functions have typically been applied for the spatial approximation. In order to improve the tractability of an orthogonal polynomial approximation in both phase-space coordinates, we propose a careful strategy for computing the propagation operator. For the favourable case of a Legendre polynomial basis we show that the integrals in the definition of the propagation operator may be evaluated analytically with respect to position and via a spectrally convergent quadrature rule for the direction coordinate. A generally applicable spectral quadrature scheme for integration with respect to both coordinates is also detailed for completeness. Finally, we provide numerical results that motivate the use of p-refinement in the orthogonal polynomial basis.

Citation

Bajars, J., Chappell, D., Hartmann, T., & Tanner, G. (2017). Improved approximation of phase-space densities on triangulated domains using discrete flow mapping with p-refinement. Journal of Scientific Computing, 72(3), https://doi.org/10.1007/s10915-017-0397-8

Journal Article Type Article
Acceptance Date Feb 18, 2017
Online Publication Date Mar 4, 2017
Publication Date Sep 30, 2017
Deposit Date Sep 21, 2017
Publicly Available Date Mar 29, 2024
Journal Journal of Scientific Computing
Print ISSN 0885-7474
Electronic ISSN 1573-7691
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 72
Issue 3
DOI https://doi.org/10.1007/s10915-017-0397-8
Keywords High frequency wave asymptotics, Ray tracing, Frobenius–Perron operator, Liouville equation, Geometrical optics, Vibro-acoustics
Public URL https://nottingham-repository.worktribe.com/output/885506
Publisher URL https://link.springer.com/article/10.1007%2Fs10915-017-0397-8

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