19 research outputs found
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
Massively Parallel Stencil Strategies for Radiation Transport Moment Model Simulations
The radiation transport equation is a mesoscopic equation in high dimensional
phase space. Moment methods approximate it via a system of partial differential
equations in traditional space-time. One challenge is the high computational
intensity due to large vector sizes (1600 components for P39) in each spatial
grid point. In this work, we extend the calculable domain size in 3D
simulations considerably, by implementing the StaRMAP methodology within the
massively parallel HPC framework NAStJA, which is designed to use current
supercomputers efficiently. We apply several optimization techniques, including
a new memory layout and explicit SIMD vectorization. We showcase a simulation
with 200 billion degrees of freedom, and argue how the implementations can be
extended and used in many scientific domains.Comment: ICCS 2020 Proceeding
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
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