Improved approximation of phase-space densities on triangulated domains using discrete flow mapping with p-refinement
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 e...
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| Format: | Article |
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Springer
2017
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| Online Access: | https://eprints.nottingham.ac.uk/46591/ |
| _version_ | 1848797362239045632 |
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| author | Bajars, Janis Chappell, David Hartmann, Timo Tanner, Gregor |
| author_facet | Bajars, Janis Chappell, David Hartmann, Timo Tanner, Gregor |
| author_sort | Bajars, Janis |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:02:40Z |
| format | Article |
| id | nottingham-46591 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:02:40Z |
| publishDate | 2017 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-465912020-05-04T19:09:41Z https://eprints.nottingham.ac.uk/46591/ Improved approximation of phase-space densities on triangulated domains using discrete flow mapping with p-refinement Bajars, Janis Chappell, David Hartmann, Timo Tanner, Gregor 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. Springer 2017-09-30 Article PeerReviewed Bajars, Janis, Chappell, David, Hartmann, Timo and Tanner, Gregor (2017) Improved approximation of phase-space densities on triangulated domains using discrete flow mapping with p-refinement. Journal of Scientific Computing, 72 (3). pp. 1290-1312. ISSN 1573-7691 High frequency wave asymptotics Ray tracing Frobenius–Perron operator Liouville equation Geometrical optics Vibro-acoustics https://link.springer.com/article/10.1007%2Fs10915-017-0397-8 doi:10.1007/s10915-017-0397-8 doi:10.1007/s10915-017-0397-8 |
| spellingShingle | High frequency wave asymptotics Ray tracing Frobenius–Perron operator Liouville equation Geometrical optics Vibro-acoustics Bajars, Janis Chappell, David Hartmann, Timo Tanner, Gregor Improved approximation of phase-space densities on triangulated domains using discrete flow mapping with p-refinement |
| title | Improved approximation of phase-space densities
on triangulated domains using discrete flow mapping with p-refinement |
| title_full | Improved approximation of phase-space densities
on triangulated domains using discrete flow mapping with p-refinement |
| title_fullStr | Improved approximation of phase-space densities
on triangulated domains using discrete flow mapping with p-refinement |
| title_full_unstemmed | Improved approximation of phase-space densities
on triangulated domains using discrete flow mapping with p-refinement |
| title_short | Improved approximation of phase-space densities
on triangulated domains using discrete flow mapping with p-refinement |
| title_sort | improved approximation of phase-space densities
on triangulated domains using discrete flow mapping with p-refinement |
| topic | High frequency wave asymptotics Ray tracing Frobenius–Perron operator Liouville equation Geometrical optics Vibro-acoustics |
| url | https://eprints.nottingham.ac.uk/46591/ https://eprints.nottingham.ac.uk/46591/ https://eprints.nottingham.ac.uk/46591/ |