Improved approximation of phasespace densities on triangulated domains using discrete flow mapping with prefinementTools Bajars, Janis and Chappell, David and Hartmann, Timo and Tanner, Gregor (2017) Improved approximation of phasespace densities on triangulated domains using discrete flow mapping with prefinement. Journal of Scientific Computing, 72 (3). pp. 12901312. ISSN 15737691
AbstractWe consider the approximation of the phasespace 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 phasespace 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 phasespace 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 prefinement in the orthogonal polynomial basis.
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