5,163 research outputs found
Ehrenfest regularization of Hamiltonian systems
Imagine a freely rotating rigid body. The body has three principal axes of
rotation. It follows from mathematical analysis of the evolution equations that
pure rotations around the major and minor axes are stable while rotation around
the middle axis is unstable. However, only rotation around the major axis (with
highest moment of inertia) is stable in physical reality (as demonstrated by
the unexpected change of rotation of the Explorer 1 probe). We propose a
general method of Ehrenfest regularization of Hamiltonian equations by which
the reversible Hamiltonian equations are equipped with irreversible terms
constructed from the Hamiltonian dynamics itself. The method is demonstrated on
harmonic oscillator, rigid body motion (solving the problem of stable minor
axis rotation), ideal fluid mechanics and kinetic theory. In particular, the
regularization can be seen as a birth of irreversibility and dissipation. In
addition, we discuss and propose discretizations of the Ehrenfest regularized
evolution equations such that key model characteristics (behavior of energy and
entropy) are valid in the numerical scheme as well
Smoothed Particle Hydrodynamics and Magnetohydrodynamics
This paper presents an overview and introduction to Smoothed Particle
Hydrodynamics and Magnetohydrodynamics in theory and in practice. Firstly, we
give a basic grounding in the fundamentals of SPH, showing how the equations of
motion and energy can be self-consistently derived from the density estimate.
We then show how to interpret these equations using the basic SPH interpolation
formulae and highlight the subtle difference in approach between SPH and other
particle methods. In doing so, we also critique several `urban myths' regarding
SPH, in particular the idea that one can simply increase the `neighbour number'
more slowly than the total number of particles in order to obtain convergence.
We also discuss the origin of numerical instabilities such as the pairing and
tensile instabilities. Finally, we give practical advice on how to resolve
three of the main issues with SPMHD: removing the tensile instability,
formulating dissipative terms for MHD shocks and enforcing the divergence
constraint on the particles, and we give the current status of developments in
this area. Accompanying the paper is the first public release of the NDSPMHD
SPH code, a 1, 2 and 3 dimensional code designed as a testbed for SPH/SPMHD
algorithms that can be used to test many of the ideas and used to run all of
the numerical examples contained in the paper.Comment: 44 pages, 14 figures, accepted to special edition of J. Comp. Phys.
on "Computational Plasma Physics". The ndspmhd code is available for download
from http://users.monash.edu.au/~dprice/ndspmhd
Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems
The purpose of this work is twofold. First, we demonstrate analytically
that the classical Newmark family as well as related integration
algorithms are variational in the sense of the Veselov formulation of
discrete mechanics. Such variational algorithms are well known to be
symplectic and momentum preserving and to often have excellent global
energy behavior. This analytical result is veried through numerical examples
and is believed to be one of the primary reasons that this class
of algorithms performs so well.
Second, we develop algorithms for mechanical systems with forcing,
and in particular, for dissipative systems. In this case, we develop integrators
that are based on a discretization of the Lagrange d'Alembert
principle as well as on a variational formulation of dissipation. It is
demonstrated that these types of structured integrators have good numerical
behavior in terms of obtaining the correct amounts by which
the energy changes over the integration run
Numerical Methods for the Stochastic Landau-Lifshitz Navier-Stokes Equations
The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal
fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This
paper examines explicit Eulerian discretizations of the full LLNS equations.
Several CFD approaches are considered (including MacCormack's two-step
Lax-Wendroff scheme and the Piecewise Parabolic Method) and are found to give
good results (about 10% error) for the variances of momentum and energy
fluctuations. However, neither of these schemes accurately reproduces the
density fluctuations. We introduce a conservative centered scheme with a
third-order Runge-Kutta temporal integrator that does accurately produce
density fluctuations. A variety of numerical tests, including the random walk
of a standing shock wave, are considered and results from the stochastic LLNS
PDE solver are compared with theory, when available, and with molecular
simulations using a Direct Simulation Monte Carlo (DSMC) algorithm
Smoothed Particle Magnetohydrodynamics III. Multidimensional tests and the div B = 0 constraint
In two previous papers (Price & Monaghan 2004a,b) (papers I,II) we have
described an algorithm for solving the equations of Magnetohydrodynamics (MHD)
using the Smoothed Particle Hydrodynamics (SPH) method. The algorithm uses
dissipative terms in order to capture shocks and has been tested on a wide
range of one dimensional problems in both adiabatic and isothermal MHD. In this
paper we investigate multidimensional aspects of the algorithm, refining many
of the aspects considered in papers I and II and paying particular attention to
the code's ability to maintain the div B = 0 constraint associated with the
magnetic field. In particular we implement a hyperbolic divergence cleaning
method recently proposed by Dedner et al. (2002) in combination with the
consistent formulation of the MHD equations in the presence of non-zero
magnetic divergence derived in papers I and II. Various projection methods for
maintaining the divergence-free condition are also examined. Finally the
algorithm is tested against a wide range of multidimensional problems used to
test recent grid-based MHD codes. A particular finding of these tests is that
in SPMHD the magnitude of the divergence error is dependent on the number of
neighbours used to calculate a particle's properties and only weakly dependent
on the total number of particles. Whilst many improvements could still be made
to the algorithm, our results suggest that the method is ripe for application
to problems of current theoretical interest, such as that of star formation.Comment: Here is the latest offering in my quest for a decent SPMHD algorithm.
26 pages, 15 figures, accepted for publication in MNRAS. Version with high
res figures available from
http://www.astro.ex.ac.uk/people/dprice/pubs/spmhd/spmhdpaper3.pd
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