4,408 research outputs found
A New Spherical Harmonics Scheme for Multi-Dimensional Radiation Transport I: Static Matter Configurations
Recent work by McClarren & Hauck [29] suggests that the filtered spherical
harmonics method represents an efficient, robust, and accurate method for
radiation transport, at least in the two-dimensional (2D) case. We extend their
work to the three-dimensional (3D) case and find that all of the advantages of
the filtering approach identified in 2D are present also in the 3D case. We
reformulate the filter operation in a way that is independent of the timestep
and of the spatial discretization. We also explore different second- and
fourth-order filters and find that the second-order ones yield significantly
better results. Overall, our findings suggest that the filtered spherical
harmonics approach represents a very promising method for 3D radiation
transport calculations.Comment: 29 pages, 13 figures. Version matching the one in Journal of
Computational Physic
A class of Galerkin schemes for time-dependent radiative transfer
The numerical solution of time-dependent radiative transfer problems is
challenging, both, due to the high dimension as well as the anisotropic
structure of the underlying integro-partial differential equation. In this
paper we propose a general framework for designing numerical methods for
time-dependent radiative transfer based on a Galerkin discretization in space
and angle combined with appropriate time stepping schemes. This allows us to
systematically incorporate boundary conditions and to preserve basic properties
like exponential stability and decay to equilibrium also on the discrete level.
We present the basic a-priori error analysis and provide abstract error
estimates that cover a wide class of methods. The starting point for our
considerations is to rewrite the radiative transfer problem as a system of
evolution equations which has a similar structure like first order hyperbolic
systems in acoustics or electrodynamics. This analogy allows us to generalize
the main arguments of the numerical analysis for such applications to the
radiative transfer problem under investigation. We also discuss a particular
discretization scheme based on a truncated spherical harmonic expansion in
angle, a finite element discretization in space, and the implicit Euler method
in time. The performance of the resulting mixed PN-finite element time stepping
scheme is demonstrated by computational results
Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering
In highly diffusion regimes when the mean free path tends to
zero, the radiative transfer equation has an asymptotic behavior which is
governed by a diffusion equation and the corresponding boundary condition.
Generally, a numerical scheme for solving this problem has the truncation error
containing an contribution, that leads to a nonuniform
convergence for small . Such phenomenons require high resolutions
of discretizations, which degrades the performance of the numerical scheme in
the diffusion limit. In this paper, we first provide a--priori estimates for
the scaled spherical harmonic () radiative transfer equation. Then we
present an error analysis for the spherical harmonic discontinuous Galerkin
(DG) method of the scaled radiative transfer equation showing that, under some
mild assumptions, its solutions converge uniformly in to the
solution of the scaled radiative transfer equation. We further present an
optimal convergence result for the DG method with the upwind flux on Cartesian
grids. Error estimates of
(where is the maximum element length) are obtained when tensor product
polynomials of degree at most are used
Incorporating reflection boundary conditions in the Neumann series radiative transport equation: Application to photon propagation and reconstruction in diffuse optical imaging
We propose a formalism to incorporate boundary conditions in a Neumann-series-based radiative transport equation. The formalism accurately models the reflection of photons at the tissue-external medium interface using Fresnelâs equations. The formalism was used to develop a gradient descent-based image reconstruction technique. The proposed methods were implemented for 3D diffuse optical imaging. In computational studies, it was observed that the average root-mean-square error (RMSE) for the output images and the estimated absorption coefficients reduced by 38% and 84%, respectively, when the reflection boundary conditions were incorporated. These results demonstrate the importance of incorporating boundary conditions that model the reflection of photons at the tissue-external medium interface
An algorithm for computing the 2D structure of fast rotating stars
Stars may be understood as self-gravitating masses of a compressible fluid
whose radiative cooling is compensated by nuclear reactions or gravitational
contraction. The understanding of their time evolution requires the use of
detailed models that account for a complex microphysics including that of
opacities, equation of state and nuclear reactions. The present stellar models
are essentially one-dimensional, namely spherically symmetric. However, the
interpretation of recent data like the surface abundances of elements or the
distribution of internal rotation have reached the limits of validity of
one-dimensional models because of their very simplified representation of
large-scale fluid flows. In this article, we describe the ESTER code, which is
the first code able to compute in a consistent way a two-dimensional model of a
fast rotating star including its large-scale flows. Compared to classical 1D
stellar evolution codes, many numerical innovations have been introduced to
deal with this complex problem. First, the spectral discretization based on
spherical harmonics and Chebyshev polynomials is used to represent the 2D
axisymmetric fields. A nonlinear mapping maps the spheroidal star and allows a
smooth spectral representation of the fields. The properties of Picard and
Newton iterations for solving the nonlinear partial differential equations of
the problem are discussed. It turns out that the Picard scheme is efficient on
the computation of the simple polytropic stars, but Newton algorithm is
unsurpassed when stellar models include complex microphysics. Finally, we
discuss the numerical efficiency of our solver of Newton iterations. This
linear solver combines the iterative Conjugate Gradient Squared algorithm
together with an LU-factorization serving as a preconditionner of the Jacobian
matrix.Comment: 40 pages, 12 figures, accepted in J. Comput. Physic
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