7,507 research outputs found
Numerical methods with controlled dissipation for small-scale dependent shocks
We provide a ‘user guide' to the literature of the past twenty years concerning the modelling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admit small-scale dependent shock waves. We cover several classes of problems and solutions: nonclassical undercompressive shocks, hyperbolic systems in nonconservative form, and boundary layer problems. We review the relevant models arising in continuum physics and describe the numerical methods that have been proposed to capture small-scale dependent solutions. In agreement with general well-posedness theory, small-scale dependent solutions are characterized by a kinetic relation, a family of paths, or an admissible boundary set. We provide a review of numerical methods (front-tracking schemes, finite difference schemes, finite volume schemes), which, at the discrete level, reproduce the effect of the physically meaningful dissipation mechanisms of interest in the applications. An essential role is played by the equivalent equation associated with discrete schemes, which is found to be relevant even for solutions containing shock wave
Self-Similar Evolution of Cosmic-Ray-Modified Quasi-Parallel Plane Shocks
Using an improved version of the previously introduced CRASH (Cosmic Ray
Acceleration SHock) code, we have calculated the time evolution of cosmic-ray
(CR) modified quasi-parallel plane shocks for Bohm-like diffusion, including
self-consistent models of Alfven wave drift and dissipation, along with thermal
leakage injection of CRs. The new simulations follow evolution of the CR
distribution to much higher energies than our previous study, providing a
better examination of evolutionary and asymptotic behaviors. The postshock CR
pressure becomes constant after quick initial adjustment, since the evolution
of the CR partial pressure expressed in terms of a momentum similarity variable
is self-similar. The shock precursor, which scales as the diffusion length of
the highest energy CRs, subsequently broadens approximately linearly with time,
independent of diffusion model, so long as CRs continue to be accelerated to
ever-higher energies. This means the nonlinear shock structure can be described
approximately in terms of the similarity variable, x/(u_s t), where u_s is the
shock speed once the postshock pressure reaches an approximate time asymptotic
state. As before, the shock Mach number is the key parameter determining the
evolution and the CR acceleration efficiency, although finite Alfven wave drift
and wave energy dissipation in the shock precursor reduce the effective
velocity change experienced by CRs, so reduce acceleration efficiency
noticeably, thus, providing a second important parameter at low and moderate
Mach numbers.Comment: 29 pages, 8 figure
Effect of molecular relaxation on the propagation of sonic booms through a stratified atmosphere
Nonlinear acoustic wave propagation through a stratified atmosphere is considered. The initial signal is taken to be an isolated N-wave, which is the disturbance that is generated some distance away from a supersonic body in horizontal flight. The effect of cylindrical spreading and exponential density stratification on the propagation of the disturbance is considered, with the shock structure controlled by molecular relaxation mechanisms and by thermoviscous diffusion. An augmented Burgers equation is obtained and asymptotic solutions are derived based on the limit of small dissipation and dispersion. For a single relaxation mode, the solution depends on whether relaxation alone can support the shock or whether a sub-shock arises controlled by other mechanisms. The resulting shock structures are known as fully dispersed and partly dispersed shocks, respectively. In this paper, the spatial location of the transition between fully dispersed and partly dispersed shocks is identified for shocks propagating above and below the horizontal. This phenomenon is important in understanding the character of sonic booms since the transition to a partly dispersed shock structure leads to the appearance of a shorter scale in the shock rise-time, associated with the embedded sub-shock
Accurate, Meshless Methods for Magneto-Hydrodynamics
Recently, we developed a pair of meshless finite-volume Lagrangian methods
for hydrodynamics: the 'meshless finite mass' (MFM) and 'meshless finite
volume' (MFV) methods. These capture advantages of both smoothed-particle
hydrodynamics (SPH) and adaptive mesh-refinement (AMR) schemes. Here, we extend
these to include ideal magneto-hydrodynamics (MHD). The MHD equations are
second-order consistent and conservative. We augment these with a
divergence-cleaning scheme, which maintains div*B~0 to high accuracy. We
implement these in the code GIZMO, together with a state-of-the-art
implementation of SPH MHD. In every one of a large suite of test problems, the
new methods are competitive with moving-mesh and AMR schemes using constrained
transport (CT) to ensure div*B=0. They are able to correctly capture the growth
and structure of the magneto-rotational instability (MRI), MHD turbulence, and
the launching of magnetic jets, in some cases converging more rapidly than AMR
codes. Compared to SPH, the MFM/MFV methods exhibit proper convergence at fixed
neighbor number, sharper shock capturing, and dramatically reduced noise, div*B
errors, and diffusion. Still, 'modern' SPH is able to handle most of our tests,
at the cost of much larger kernels and 'by hand' adjustment of artificial
diffusion parameters. Compared to AMR, the new meshless methods exhibit
enhanced 'grid noise' but reduced advection errors and numerical diffusion,
velocity-independent errors, and superior angular momentum conservation and
coupling to N-body gravity solvers. As a result they converge more slowly on
some problems (involving smooth, slowly-moving flows) but more rapidly on
others (involving advection or rotation). In all cases, divergence-control
beyond the popular Powell 8-wave approach is necessary, or else all methods we
consider will systematically converge to unphysical solutions.Comment: 35 pages, 39 figures. MNRAS. Updated with published version. A public
version of the GIZMO MHD code, user's guide, test problem setups, and movies
are available at http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.htm
On the convergence of the critical cooling timescale for the fragmentation of self-gravitating discs
We carry out simulations of gravitationally unstable discs using a Smoothed
Particle Hydrodynamics (SPH) code and a grid-based hydrodynamics code, FARGO,
to understand the previous non-convergent results reported by Meru & Bate
(2011a). We obtain evidence that convergence with increasing resolution occurs
with both SPH and FARGO and in both cases we find that the critical cooling
timescale is larger than previously thought. We show that SPH has a first-order
convergence rate while FARGO converges with a second-order rate. We show that
the convergence of the critical cooling timescale for fragmentation depends
largely on the numerical viscosity employed in both SPH and FARGO. With SPH,
particle velocity dispersion may also play a role. We show that reducing the
dissipation from the numerical viscosity leads to larger values of the critical
cooling time at a given resolution. For SPH, we find that the effect of the
dissipation due to the numerical viscosity is somewhat larger than had
previously been appreciated. In particular, we show that using a quadratic term
in the SPH artificial viscosity (beta_{SPH}) that is too low appears to lead to
excess dissipation in gravitationally unstable discs, which may affect any
results that sensitively depend on the thermodynamics, such as disc
fragmentation. We show that the two codes converge to values of the critical
cooling timescale, beta_{crit} > 20 (for a ratio of specific heats of
gamma=5/3), and perhaps even as large as beta_{crit} \approx 30. These are
approximately 3-5 times larger than has been found by most previous studies.
This is equivalent to a maximum gravitational stress that a disc can withstand
without fragmenting of alpha_{GI,crit} \approx 0.013-0.02, which is much
smaller than the values typically used in the literature. It is therefore
easier for self-gravitating discs to fragment than has been concluded from most
past studies.Comment: Accepted for publication by MNRAS. 26 pages, 17 figure
Subsonic turbulence in smoothed particle hydrodynamics and moving-mesh simulations
Highly supersonic, compressible turbulence is thought to be of tantamount
importance for star formation processes in the interstellar medium. Likewise,
cosmic structure formation is expected to give rise to subsonic turbulence in
the intergalactic medium, which may substantially modify the thermodynamic
structure of gas in virialized dark matter halos and affect small-scale mixing
processes in the gas. Numerical simulations have played a key role in
characterizing the properties of astrophysical turbulence, but thus far
systematic code comparisons have been restricted to the supersonic regime,
leaving it unclear whether subsonic turbulence is faithfully represented by the
numerical techniques commonly employed in astrophysics. Here we focus on
comparing the accuracy of smoothed particle hydrodynamics (SPH) and our new
moving-mesh technique AREPO in simulations of driven subsonic turbulence. To
make contact with previous results, we also analyze simulations of transsonic
and highly supersonic turbulence. We find that the widely employed standard
formulation of SPH yields problematic results in the subsonic regime. Instead
of building up a Kolmogorov-like turbulent cascade, large-scale eddies are
quickly damped close to the driving scale and decay into small-scale velocity
noise. Reduced viscosity settings improve the situation, but the shape of the
dissipation range differs compared with expectations for a Kolmogorov cascade.
In contrast, our moving-mesh technique does yield power-law scaling laws for
the power spectra of velocity, vorticity and density, consistent with
expectations for fully developed isotropic turbulence. We show that large
errors in SPH's gradient estimate and the associated subsonic velocity noise
are ultimately responsible for producing inaccurate results in the subsonic
regime. In contrast, SPH's performance is much better for supersonic
turbulence. [Abridged]Comment: 22 pages, 20 figures, accepted in MNRAS. Includes a rebuttal to
arXiv:1111.1255 of D. Price and significant revisions to address referee
comments. Conclusions of original submission unchange
Numerical simulation of the three-dimensional structure and dynamics of the non-magnetic solar chromosphere
Three-dimensional numerical simulations with CO5BOLD, a new radiation
hydrodynamics code, result in a dynamic, thermally bifurcated model of the
non-magnetic chromosphere of the quiet Sun. The 3-D model includes the middle
and low chromosphere, the photosphere, and the top of the convection zone,
where acoustic waves are excited by convective motions. While the waves
propagate upwards, they steepen into shocks, dissipate, and deposit their
mechanical energy as heat in the chromosphere. Our numerical simulations show
for the first time a complex 3-D structure of the chromospheric layers, formed
by the interaction of shock waves. Horizontal temperature cross-sections of the
model chromosphere exhibit a network of hot filaments and enclosed cool
regions. The horizontal pattern evolves on short time-scales of the order of
typically 20 - 25 seconds, and has spatial scales comparable to those of the
underlying granulation. The resulting thermal bifurcation, i.e., the
co-existence of cold and hot regions, provides temperatures high enough to
produce the observed chromospheric UV emission and -- at the same time --
temperatures cold enough to allow the formation of molecules (e.g., carbon
monoxide). Our 3-D model corroborates the finding by Carlsson & Stein (1994)
that the chromospheric temperature rise of semi-empirical models does not
necessarily imply an increase in the average gas temperature but can be
explained by the presence of substantial spatial and temporal temperature
inhomogeneities.Comment: 18 pages, 13 figures, accepted by Astronomy & Astrophysics (30/10/03
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