7,666 research outputs found
Optimal prediction for moment models: Crescendo diffusion and reordered equations
A direct numerical solution of the radiative transfer equation or any kinetic
equation is typically expensive, since the radiative intensity depends on time,
space and direction. An expansion in the direction variables yields an
equivalent system of infinitely many moments. A fundamental problem is how to
truncate the system. Various closures have been presented in the literature. We
want to study moment closure generally within the framework of optimal
prediction, a strategy to approximate the mean solution of a large system by a
smaller system, for radiation moment systems. We apply this strategy to
radiative transfer and show that several closures can be re-derived within this
framework, e.g. , diffusion, and diffusion correction closures. In
addition, the formalism gives rise to new parabolic systems, the reordered
equations, that are similar to the simplified equations.
Furthermore, we propose a modification to existing closures. Although simple
and with no extra cost, this newly derived crescendo diffusion yields better
approximations in numerical tests.Comment: Revised version: 17 pages, 6 figures, presented at Workshop on Moment
Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor
correction
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
StaRMAP - A second order staggered grid method for spherical harmonics moment equations of radiative transfer
We present a simple method to solve spherical harmonics moment systems, such
as the the time-dependent and equations, of radiative transfer.
The method, which works for arbitrary moment order , makes use of the
specific coupling between the moments in the equations. This coupling
naturally induces staggered grids in space and time, which in turn give rise to
a canonical, second-order accurate finite difference scheme. While the scheme
does not possess TVD or realizability limiters, its simplicity allows for a
very efficient implementation in Matlab. We present several test cases, some of
which demonstrate that the code solves problems with ten million degrees of
freedom in space, angle, and time within a few seconds. The code for the
numerical scheme, called StaRMAP (Staggered grid Radiation Moment
Approximation), along with files for all presented test cases, can be
downloaded so that all results can be reproduced by the reader.Comment: 28 pages, 7 figures; StaRMAP code available at
http://www.math.temple.edu/~seibold/research/starma
A Two-moment Radiation Hydrodynamics Module in Athena Using a Time-explicit Godunov Method
We describe a module for the Athena code that solves the gray equations of
radiation hydrodynamics (RHD), based on the first two moments of the radiative
transfer equation. We use a combination of explicit Godunov methods to advance
the gas and radiation variables including the non-stiff source terms, and a
local implicit method to integrate the stiff source terms. We adopt the M1
closure relation and include all leading source terms. We employ the reduced
speed of light approximation (RSLA) with subcycling of the radiation variables
in order to reduce computational costs. Our code is dimensionally unsplit in
one, two, and three space dimensions and is parallelized using MPI. The
streaming and diffusion limits are well-described by the M1 closure model, and
our implementation shows excellent behavior for a problem with a concentrated
radiation source containing both regimes simultaneously. Our operator-split
method is ideally suited for problems with a slowly varying radiation field and
dynamical gas flows, in which the effect of the RSLA is minimal. We present an
analysis of the dispersion relation of RHD linear waves highlighting the
conditions of applicability for the RSLA. To demonstrate the accuracy of our
method, we utilize a suite of radiation and RHD tests covering a broad range of
regimes, including RHD waves, shocks, and equilibria, which show second-order
convergence in most cases. As an application, we investigate radiation-driven
ejection of a dusty, optically thick shell in the interstellar medium (ISM).
Finally, we compare the timing of our method with other well-known iterative
schemes for the RHD equations. Our code implementation, Hyperion, is suitable
for a wide variety of astrophysical applications and will be made freely
available on the Web.Comment: 30 pages, 29 figures, accepted for publication in ApJ
Numerical Solution of the Expanding Stellar Atmosphere Problem
In this paper we discuss numerical methods and algorithms for the solution of
NLTE stellar atmosphere problems involving expanding atmospheres, e.g., found
in novae, supernovae and stellar winds. We show how a scheme of nested
iterations can be used to reduce the high dimension of the problem to a number
of problems with smaller dimensions. As examples of these sub-problems, we
discuss the numerical solution of the radiative transfer equation for
relativistically expanding media with spherical symmetry, the solution of the
multi-level non-LTE statistical equilibrium problem for extremely large model
atoms, and our temperature correction procedure. Although modern iteration
schemes are very efficient, parallel algorithms are essential in making large
scale calculations feasible, therefore we discuss some parallelization schemes
that we have developed.Comment: JCAM, in press. 28 pages, also available at
ftp://calvin.physast.uga.edu:/pub/preprints/CompAstro.ps.g
Second-order mixed-moment model with differentiable ansatz function in slab geometry
We study differentiable mixed-moment models (full zeroth and first moment,
half higher moments) for a Fokker-Planck equation in one space dimension.
Mixed-moment minimum-entropy models are known to overcome the zero net-flux
problem of full-moment minimum entropy models. Realizability theory for
these modification of mixed moments is derived for second order. Numerical
tests are performed with a kinetic first-order finite volume scheme and
compared with , classical and a reference scheme.Comment: arXiv admin note: text overlap with arXiv:1611.01314,
arXiv:1511.0271
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