10,360 research outputs found
An Unsplit Godunov Method for Ideal MHD via Constrained Transport in Three Dimensions
We present a single step, second-order accurate Godunov scheme for ideal MHD
which is an extension of the method described by Gardiner & Stone (2005) to
three dimensions. This algorithm combines the corner transport upwind (CTU)
method of Colella for multidimensional integration, and the constrained
transport (CT) algorithm for preserving the divergence-free constraint on the
magnetic field. We describe the calculation of the PPM interface states for 3D
ideal MHD which must include multidimensional ``MHD source terms'' and
naturally respect the balance implicit in these terms by the condition. We compare two different forms for the CTU integration
algorithm which require either 6- or 12-solutions of the Riemann problem per
cell per time-step, and present a detailed description of the 6-solve
algorithm. Finally, we present solutions for test problems to demonstrate the
accuracy and robustness of the algorithm.Comment: Extended version of the paper accepted for publication in JC
Sub-Alfvenic Non-Ideal MHD Turbulence Simulations with Ambipolar Diffusion: I. Turbulence Statistics
Most numerical investigations on the role of magnetic fields in turbulent
molecular clouds (MCs) are based on ideal magneto-hydrodynamics (MHD). However,
MCs are weakly ionized, so that the time scale required for the magnetic field
to diffuse through the neutral component of the plasma by ambipolar diffusion
(AD) can be comparable to the dynamical time scale. We have performed a series
of 256^3 and 512^3 simulations on supersonic but sub-Alfvenic turbulent systems
with AD using the Heavy-Ion Approximation developed in Li, McKee, & Klein
(2006). Our calculations are based on the assumption that the number of ions is
conserved, but we show that these results approximately apply to the case of
time-dependent ionization in molecular clouds as well. Convergence studies
allow us to determine the optimal value of the ionization mass fraction when
using the heavy-ion approximation for low Mach number, sub-Alfvenic turbulent
systems. We find that ambipolar diffusion steepens the velocity and magnetic
power spectra compared to the ideal MHD case. Changes in the density PDF, total
magnetic energy, and ionization fraction are determined as a function of the AD
Reynolds number. The power spectra for the neutral gas properties of a strongly
magnetized medium with a low AD Reynolds number are similar to those for a
weakly magnetized medium; in particular, the power spectrum of the neutral
velocity is close to that for Burgers turbulence.Comment: 37 pages, 11 figures, 4 table
Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code
In this paper we present a rigorous derivation of the reduced MHD models with
and without parallel velocity that are implemented in the non-linear MHD code
JOREK. The model we obtain contains some terms that have been neglected in the
implementation but might be relevant in the non-linear phase. These are
necessary to guarantee exact conservation with respect to the full MHD energy.
For the second part of this work, we have replaced the linearized time stepping
of JOREK by a non-linear solver based on the Inexact Newton method including
adaptive time stepping. We demonstrate that this approach is more robust
especially with respect to numerical errors in the saturation phase of an
instability and allows to use larger time steps in the non-linear phase
General-relativistic resistive magnetohydrodynamics in three dimensions: Formulation and tests
We present a new numerical implementation of the general-relativistic
resistive magnetohydrodynamics (MHD) equations within the Whisky code. The
numerical method adopted exploits the properties of implicit-explicit
Runge-Kutta numerical schemes to treat the stiff terms that appear in the
equations for large electrical conductivities. Using tests in one, two, and
three dimensions, we show that our implementation is robust and recovers the
ideal-MHD limit in regimes of very high conductivity. Moreover, the results
illustrate that the code is capable of describing scenarios in a very wide
range of conductivities. In addition to tests in flat spacetime, we report
simulations of magnetized nonrotating relativistic stars, both in the Cowling
approximation and in dynamical spacetimes. Finally, because of its
astrophysical relevance and because it provides a severe testbed for
general-relativistic codes with dynamical electromagnetic fields, we study the
collapse of a nonrotating star to a black hole. We show that also in this case
our results on the quasinormal mode frequencies of the excited electromagnetic
fields in the Schwarzschild background agree with the perturbative studies
within 0.7% and 5.6% for the real and the imaginary part of the l=1 mode
eigenfrequency, respectively. Finally we provide an estimate of the
electromagnetic efficiency of this process.Comment: 22 pages, 19 figure
An Unsplit, Cell-Centered Godunov Method for Ideal MHD
We present a second-order Godunov algorithm for multidimensional, ideal MHD.
Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys.
vol. 87, 1990), with all of the primary dependent variables centered at the
same location. To properly represent the divergence-free condition of the
magnetic fields, we apply a discrete projection to the intermediate values of
the field at cell faces, and apply a filter to the primary dependent variables
at the end of each time step. We test the method against a suite of linear and
nonlinear tests to ascertain accuracy and stability of the scheme under a
variety of conditions. The test suite includes rotated planar linear waves, MHD
shock tube problems, low-beta flux tubes, and a magnetized rotor problem. For
all of these cases, we observe that the algorithm is second-order accurate for
smooth solutions, converges to the correct weak solution for problems involving
shocks, and exhibits no evidence of instability or loss of accuracy due to the
possible presence of non-solenoidal fields.Comment: 37 Pages, 9 Figures, submitted to Journal of Computational Physic
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