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

Bajars, Janis and Chappell, David and 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

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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.

Item Type: Article
Keywords: High frequency wave asymptotics, Ray tracing, Frobenius–Perron operator, Liouville equation, Geometrical optics, Vibro-acoustics
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: 10.1007/s10915-017-0397-8
Depositing User: Tanner, Prof Gregor
Date Deposited: 21 Sep 2017 13:49
Last Modified: 14 Oct 2017 07:40
URI: http://eprints.nottingham.ac.uk/id/eprint/46591

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