7,015 research outputs found
A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method
In this note, two numerical methods of solving fractional differential
equations (FDEs) are briefly described, namely predictor-corrector approach of
Adams-Bashforth-Moulton type and multi-step generalized differential transform
method (MSGDTM), and then a demonstrating example is given to compare the
results of the methods. It is shown that the MSGDTM, which is an enhancement of
the generalized differential transform method, neglects the effect of non-local
structure of fractional differentiation operators and fails to accurately solve
the FDEs over large domains.Comment: 12 pages, 2 figure
An implementation of radiative transfer in the cosmological simulation code GADGET
We present a novel numerical implementation of radiative transfer in the
cosmological smoothed particle hydrodynamics (SPH) simulation code {\small
GADGET}. It is based on a fast, robust and photon-conserving integration scheme
where the radiation transport problem is approximated in terms of moments of
the transfer equation and by using a variable Eddington tensor as a closure
relation, following the `OTVET'-suggestion of Gnedin & Abel. We derive a
suitable anisotropic diffusion operator for use in the SPH discretization of
the local photon transport, and we combine this with an implicit solver that
guarantees robustness and photon conservation. This entails a matrix inversion
problem of a huge, sparsely populated matrix that is distributed in memory in
our parallel code. We solve this task iteratively with a conjugate gradient
scheme. Finally, to model photon sink processes we consider ionisation and
recombination processes of hydrogen, which is represented with a chemical
network that is evolved with an implicit time integration scheme. We present
several tests of our implementation, including single and multiple sources in
static uniform density fields with and without temperature evolution, shadowing
by a dense clump, and multiple sources in a static cosmological density field.
All tests agree quite well with analytical computations or with predictions
from other radiative transfer codes, except for shadowing. However, unlike most
other radiative transfer codes presently in use for studying reionisation, our
new method can be used on-the-fly during dynamical cosmological simulation,
allowing simultaneous treatments of galaxy formation and the reionisation
process of the Universe.Comment: 21 pages, 17 figures, published in MNRA
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