6,900 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
Asymptotic Derivation and Numerical Investigation of Time-Dependent Simplified Pn Equations
The steady-state simplified Pn (SPn) approximations to the linear Boltzmann
equation have been proven to be asymptotically higher-order corrections to the
diffusion equation in certain physical systems. In this paper, we present an
asymptotic analysis for the time-dependent simplified Pn equations up to n = 3.
Additionally, SPn equations of arbitrary order are derived in an ad hoc way.
The resulting SPn equations are hyperbolic and differ from those investigated
in a previous work by some of the authors. In two space dimensions, numerical
calculations for the Pn and SPn equations are performed. We simulate neutron
distributions of a moving rod and present results for a benchmark problem,
known as the checkerboard problem. The SPn equations are demonstrated to yield
significantly more accurate results than diffusion approximations. In addition,
for sufficiently low values of n, they are shown to be more efficient than Pn
models of comparable cost.Comment: 32 pages, 7 figure
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 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
Radiative Heat Transfer and Effective Transport Coefficients
The theory of heat transfer by electromagnetic radiation is based on the
radiative transfer equation (RTE) for the radiation intensity, or equivalently
on the Boltzmann transport equation (BTE) for the photon distribution. We focus
in this review article, after a brief overview on different solution methods,
on a recently introduced approach based on truncated moment expansion. Due to
the linearity of the underlying BTE, the appropriate closure of the system of
moment equations is entropy production rate minimization. This closure provides
a distribution function and the associated effective transport coefficients,
like mean absorption coefficients and the Eddington factor, for an arbitrary
number of moments. The moment approach is finally illustrated with an
application of the two-moment equations to an electrical arc
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