551 research outputs found
A Simple and Accurate Riemann Solver for Isothermal MHD
A new approximate Riemann solver for the equations of magnetohydrodynamics
(MHD) with an isothermal equation of state is presented.
The proposed method of solution draws on the recent work of
Miyoshi and Kusano, in the context of adiabatic MHD, where an approximate
solution to the Riemann problem is sought in terms of an average constant
velocity and total pressure across the Riemann fan.
This allows the formation of four intermediate states enclosed by two
outermost fast discontinuities and separated by two rotational waves and an
entropy mode.
In the present work, a corresponding derivation for the isothermal
MHD equations is presented.
It is found that the absence of the entropy mode leads to a different
formulation which is based on a three-state representation rather than four.
Numerical tests in one and two dimensions demonstrates that the new solver is
robust and comparable in accuracy to the more expensive linearized solver of
Roe, although considerably faster.Comment: 19 pages, 9 figure
Athena: A New Code for Astrophysical MHD
A new code for astrophysical magnetohydrodynamics (MHD) is described. The
code has been designed to be easily extensible for use with static and adaptive
mesh refinement. It combines higher-order Godunov methods with the constrained
transport (CT) technique to enforce the divergence-free constraint on the
magnetic field. Discretization is based on cell-centered volume-averages for
mass, momentum, and energy, and face-centered area-averages for the magnetic
field. Novel features of the algorithm include (1) a consistent framework for
computing the time- and edge-averaged electric fields used by CT to evolve the
magnetic field from the time- and area-averaged Godunov fluxes, (2) the
extension to MHD of spatial reconstruction schemes that involve a
dimensionally-split time advance, and (3) the extension to MHD of two different
dimensionally-unsplit integration methods. Implementation of the algorithm in
both C and Fortran95 is detailed, including strategies for parallelization
using domain decomposition. Results from a test suite which includes problems
in one-, two-, and three-dimensions for both hydrodynamics and MHD are given,
not only to demonstrate the fidelity of the algorithms, but also to enable
comparisons to other methods. The source code is freely available for download
on the web.Comment: 61 pages, 36 figures. accepted by ApJ
A Multi-dimensional Code for Isothermal Magnetohydrodynamic Flows in Astrophysics
We present a multi-dimensional numerical code to solve isothermal
magnetohydrodynamic (IMHD) equations for use in modeling astrophysical flows.
First, we have built a one-dimensional code which is based on an explicit
finite-difference method on an Eulerian grid, called the total variation
diminishing (TVD) scheme. Recipes for building the one-dimensional IMHD code,
including the normalized right and left eigenvectors of the IMHD Jacobian
matrix, are presented. Then, we have extended the one-dimensional code to a
multi-dimensional IMHD code through a Strang-type dimensional splitting. In the
multi-dimensional code, an explicit cleaning step has been included to
eliminate non-zero at every time step. To estimate the
proformance of the code, one- and two-dimensional IMHD shock tube tests, and
the decay test of a two-dimensional Alfv\'{e}n wave have been done. As an
example of astrophysical applications, we have simulated the nonlinear
evolution of the two-dimensional Parker instability under a uniform gravity.Comment: Accepted for publication in ApJ, using aaspp4.sty, 22 text pages with
10 figure
Comparing Numerical Methods for Isothermal Magnetized Supersonic Turbulence
We employ simulations of supersonic super-Alfvenic turbulence decay as a
benchmark test problem to assess and compare the performance of nine
astrophysical MHD methods actively used to model star formation. The set of
nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER,
and ZEUS. We present a comprehensive set of statistical measures designed to
quantify the effects of numerical dissipation in these MHD solvers. We compare
power spectra for basic fields to determine the effective spectral bandwidth of
the methods and rank them based on their relative effective Reynolds numbers.
We also compare numerical dissipation for solenoidal and dilatational velocity
components to check for possible impacts of the numerics on small-scale density
statistics. Finally, we discuss convergence of various characteristics for the
turbulence decay test and impacts of various components of numerical schemes on
the accuracy of solutions. We show that the best performing codes employ a
consistently high order of accuracy for spatial reconstruction of the evolved
fields, transverse gradient interpolation, conservation law update step, and
Lorentz force computation. The best results are achieved with divergence-free
evolution of the magnetic field using the constrained transport method, and
using little to no explicit artificial viscosity. Codes which fall short in one
or more of these areas are still useful, but they must compensate higher
numerical dissipation with higher numerical resolution. This paper is the
largest, most comprehensive MHD code comparison on an application-like test
problem to date. We hope this work will help developers improve their numerical
algorithms while helping users to make informed choices in picking optimal
applications for their specific astrophysical problems.Comment: 17 pages, 5 color figures, revised version to appear in ApJ, 735,
July 201
Piecewise Parabolic Method on a Local Stencil for Magnetized Supersonic Turbulence Simulation
Stable, accurate, divergence-free simulation of magnetized supersonic
turbulence is a severe test of numerical MHD schemes and has been surprisingly
difficult to achieve due to the range of flow conditions present. Here we
present a new, higher order-accurate, low dissipation numerical method which
requires no additional dissipation or local "fixes" for stable execution. We
describe PPML, a local stencil variant of the popular PPM algorithm for solving
the equations of compressible ideal magnetohydrodynamics. The principal
difference between PPML and PPM is that cell interface states are evolved
rather that reconstructed at every timestep, resulting in a compact stencil.
Interface states are evolved using Riemann invariants containing all transverse
derivative information. The conservation laws are updated in an unsplit
fashion, making the scheme fully multidimensional. Divergence-free evolution of
the magnetic field is maintained using the higher order-accurate constrained
transport technique of Gardiner and Stone. The accuracy and stability of the
scheme is documented against a bank of standard test problems drawn from the
literature. The method is applied to numerical simulation of supersonic MHD
turbulence, which is important for many problems in astrophysics, including
star formation in dark molecular clouds. PPML accurately reproduces in
three-dimensions a transition to turbulence in highly compressible isothermal
gas in a molecular cloud model. The low dissipation and wide spectral bandwidth
of this method make it an ideal candidate for direct turbulence simulations.Comment: 28 pages, 18 figure
Rarefaction Shocks, Shock Errors and Low Order of Accuracy in ZEUS
We show that there are simple one dimensional problems for which the MHD
code, ZEUS, generates significant errors, whereas upwind conservative schemes
perform very well on these problems.Comment: 11 pages, accepted in ApJ Letter
Smoothed Particle Magnetohydrodynamics (some shocking results...)
There have been some issues in the past in attempts to simulate magnetic
fields using the Smoothed Particle Hydrodynamics (SPH) method. SPH is well
suited to star formation problems because of its Lagrangian nature. We present
new, stable and conservative methods for magnetohydrodynamics (MHD) in SPH and
present numerical tests on both waves and shocks in one dimension to show that
it gives robust and accurate results.Comment: Kluwer latex, 6 pages, 3 figures; Proceedings of the International
Workshop "Magnetic Fields and Star Formation: Theory vs Observations",
Madrid, 21-25 April 2003. Revised version accepted to proceedings (exact
solutions added, other minor changes
Historical perspective on astrophysical MHD simulations
This contribution contains the introductory remarks that I presented at IAU
Symposium 270 on ``Computational Star Formation" held in Barcelona, Spain, May
31 -- June 4, 2010. I discuss the historical development of numerical MHD
methods in astrophysics from a personal perspective. The recent advent of
robust, higher order-accurate MHD algorithms and adaptive mesh refinement
numerical simulations promises to greatly improve our understanding of the role
of magnetic fields in star formation.Comment: 11 pages, 5 figures, in "Computational Star Formation" held in
Barcelona, Spain, May 31 - June 4, 2010", Eds. J. Alves, B. G. Elmegreen, J.
M. Girart, V. Trimbl
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