108 research outputs found
Efficient Parallel Simulation of Atherosclerotic Plaque Formation Using Higher Order Discontinuous Galerkin Schemes
Abstract The compact Discontinuous Galerkin 2 (CDG2) method was successfully tested for elliptic problems, scalar convection-diffusion equations and compressible Navier-Stokes equations. In this paper we use the newly developed DG method to solve a mathematical model for early stages of atherosclerotic plaque formation. Atherosclerotic plaque is mainly formed by accumulation of lipid-laden cells in the arterial walls which leads to a heart attack in case the artery is occluded or a thrombus is built through a rupture of the plaque. After describing a mathematical model and the discretization scheme, we present some benchmark tests comparing the CDG2 method to other commonly used DG methods. Furthermore, we take parallelization and higher order discretization schemes into account.
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
General Relativistic Magnetohydrodynamic Bondi--Hoyle Accretion
In this paper we present a fully relativistic study of axisymmetric
magnetohydrodynamic Bondi--Hoyle accretion onto a moving Kerr black hole. The
equations of general relativistic magnetohydrodynamics are solved using high
resolution shock capturing methods. In this treatment we consider the ideal MHD
limit. The parameters of interest in this study are the adiabatic constant
, the asymptotic speed of sound , and the plasma beta
parameter . We focus the investigation on the parameter regime in
which the flow is supersonic, or when . In some
cases, subsonic asymptotic flows are considered for comparison purposes. We
study the accretion rates of the total energy and momenta, as well as the
hydrodynamic energy and momentum accretion rates. The models presented in this
study exhibit a matter density depletion in the downstream region of the black
hole which tends to vacuum in convergence tests. This feature is
due to the presence of the magnetic field, more specifically the magnetic
pressure, and is not seen in previous purely hydrodynamic studies.Comment: Version 2: The figures have been reformatted to fit the paper. All
verbal content remains identical to version
An Unstaggered Constrained Transport Method for the 3D Ideal Magnetohydrodynamic Equations
Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations
in more than one space dimension must either confront the challenge of
controlling errors in the discrete divergence of the magnetic field, or else be
faced with nonlinear numerical instabilities. One approach for controlling the
discrete divergence is through a so-called constrained transport method, which
is based on first predicting a magnetic field through a standard finite volume
solver, and then correcting this field through the appropriate use of a
magnetic vector potential. In this work we develop a constrained transport
method for the 3D ideal MHD equations that is based on a high-resolution wave
propagation scheme. Our proposed scheme is the 3D extension of the 2D scheme
developed by Rossmanith [SIAM J. Sci. Comp. 28, 1766 (2006)], and is based on
the high-resolution wave propagation method of Langseth and LeVeque [J. Comp.
Phys. 165, 126 (2000)]. In particular, in our extension we take great care to
maintain the three most important properties of the 2D scheme: (1) all
quantities, including all components of the magnetic field and magnetic
potential, are treated as cell-centered; (2) we develop a high-resolution wave
propagation scheme for evolving the magnetic potential; and (3) we develop a
wave limiting approach that is applied during the vector potential evolution,
which controls unphysical oscillations in the magnetic field. One of the key
numerical difficulties that is novel to 3D is that the transport equation that
must be solved for the magnetic vector potential is only weakly hyperbolic. In
presenting our numerical algorithm we describe how to numerically handle this
problem of weak hyperbolicity, as well as how to choose an appropriate gauge
condition. The resulting scheme is applied to several numerical test cases.Comment: 46 pages, 12 figure
A high-order Godunov scheme for global 3D MHD accretion disks simulations. I. The linear growth regime of the magneto-rotational instability
We employ the PLUTO code for computational astrophysics to assess and compare
the validity of different numerical algorithms on simulations of the
magneto-rotational instability in 3D accretion disks. In particular we stress
on the importance of using a consistent upwind reconstruction of the
electro-motive force (EMF) when using the constrained transport (CT) method to
avoid the onset of numerical instabilities. We show that the electro-motive
force (EMF) reconstruction in the classical constrained transport (CT) method
for Godunov schemes drives a numerical instability. The well-studied linear
growth of magneto-rotational instability (MRI) is used as a benchmark for an
inter-code comparison of PLUTO and ZeusMP. We reproduce the analytical results
for linear MRI growth in 3D global MHD simulations and present a robust and
accurate Godunov code which can be used for 3D accretion disk simulations in
curvilinear coordinate systems
A divergence-cleaning scheme for cosmological SPMHD simulations
In magnetohydrodynamics (MHD), the magnetic field is evolved by the induction
equation and coupled to the gas dynamics by the Lorentz force. We perform
numerical smoothed particle magnetohydrodynamics (Spmhd) simulations and study
the influence of a numerical magnetic divergence. For instabilities arising
from divergence B related errors, we find the hyperbolic/parabolic cleaning
scheme suggested by Dedner et al. 2002 to give good results and prevent
numerical artifacts from growing. Additionally, we demonstrate that certain
current Spmhd implementations of magnetic field regularizations give rise to
unphysical instabilities in long-time simulations. We also find this effect
when employing Euler potentials (divergenceless by definition), which are not
able to follow the winding-up process of magnetic field lines properly.
Furthermore, we present cosmological simulations of galaxy cluster formation at
extremely high resolution including the evolution of magnetic fields. We show
synthetic Faraday rotation maps and derive structure functions to compare them
with observations. Comparing all the simulations with and without divergence
cleaning, we are able to confirm the results of previous simulations performed
with the standard implementation of MHD in Spmhd at normal resolution. However,
at extremely high resolution, a cleaning scheme is needed to prevent the growth
of numerical errors at small scales.Comment: 15 pages, 19 figures, submitted to MNRA
MHD simulations of jet acceleration from Keplerian accretion disks: the effects of disk resistivity
Accretion disks and astrophysical jets are used to model many active
astrophysical objects, viz., young stars, relativistic stars, and active
galactic nuclei. In this paper we present self-consistent time-dependent
simulations of supersonic jets launched from magnetized accretion disks, using
high resolution numerical techniques. In particular we study the effects of the
disk magnetic resistivity, parametrized through an alpha-prescription, in
determining the properties of the inflow-outflow system. Moreover we analyze
under which conditions steady state solutions of the type proposed in the self
similar models of Blandford and Payne can be reached and maintained in a self
consistent nonlinear stage. We use the resistive MHD FLASH code with adaptive
mesh refinement, allowing us to follow the evolution of the structure for a
time scale long enough to reach steady state. A detailed analysis of the
initial configuration state is given. We obtain the expected solutions in the
axisymmetric (2.5D) limit. Assuming a magnetic field around equipartition with
the thermal pressure of the disk, we show how the characteristics of the disk
jet system, as the ejection efficiency and the energetics, are affected by the
anomalous resistivity acting inside the disk.Comment: 20 pages, 18 figures, accepted for publication in Astronomy and
Astrophysic
The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
We propose and study discontinuous Galerkin methods for strongly degenerate
convection-diffusion equations perturbed by a fractional diffusion (L\'evy)
operator. We prove various stability estimates along with convergence results
toward properly defined (entropy) solutions of linear and nonlinear equations.
Finally, the qualitative behavior of solutions of such equations are
illustrated through numerical experiments
Modeling of Protostellar Clouds and their Observational Properties
A physical model and two-dimensional numerical method for computing the
evolution and spectra of protostellar clouds are described. The physical model
is based on a system of magneto-gasdynamical equations, including ohmic and
ambipolar diffusion, and a scheme for calculating the thermal and ionization
structure of a cloud. The dust and gas temperatures are determined during the
calculations of the thermal structure of the cloud. The results of computing
the dynamical and thermal structure of the cloud are used to model the
radiative transfer in continuum and in molecular lines. We presented the
results for clouds in hydrostatic and thermal equilibrium. The evolution of a
rotating magnetic protostellar cloud starting from a quasi-static state is also
considered. Spectral maps for optically thick lines of linear molecules are
analyzed. We have shown that the influence of the magnetic field and rotation
can lead to a redistribution of angular momentum in the cloud and the formation
of a characteristic rotational velocity structure. As a result, the
distribution of the velocity centroid of the molecular lines can acquire an
hourglass shape. We plan to use the developed program package together with a
model for the chemical evolution to interpret and model observed starless and
protostellar cores.Comment: Accepted to Astronomy Report
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