351 research outputs found
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis
of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD
equations as they are particularly useful for mathematically modeling a wide
variety of magnetized fluids. In order to be self-contained we first motivate
the physical properties of a magnetic fluid and how it should behave under the
laws of thermodynamics. Next, we introduce a mathematical model built from
hyperbolic partial differential equations (PDEs) that translate physical laws
into mathematical equations. After an overview of the continuous analysis, we
thoroughly describe the derivation of a numerical approximation of the ideal
MHD system that remains consistent to the continuous thermodynamic principles.
The derivation of the method and the theorems contained within serve as the
bulk of the review article. We demonstrate that the derived numerical
approximation retains the correct entropic properties of the continuous model
and show its applicability to a variety of standard numerical test cases for
MHD schemes. We close with our conclusions and a brief discussion on future
work in the area of entropy consistent numerical methods and the modeling of
plasmas
A Flux-Differencing Formula for Split-Form Summation By Parts Discretizations of Non-Conservative Systems: Applications to Subcell Limiting for magneto-hydrodynamics
In this paper, we show that diagonal-norm summation by parts (SBP)
discretizations of general non-conservative systems of hyperbolic balance laws
can be rewritten as a finite-volume-type formula, also known as
flux-differencing formula, if the non-conservative terms can be written as the
product of a local and a symmetric contribution. Furthermore, we show that the
existence of a flux-differencing formula enables the use of recent subcell
limiting strategies to improve the robustness of the high-order
discretizations.
To demonstrate the utility of the novel flux-differencing formula, we
construct hybrid schemes that combine high-order SBP methods (the discontinuous
Galerkin spectral element method and a high-order SBP finite difference method)
with a compatible low-order finite volume (FV) scheme at the subcell level. We
apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD)
problems featuring strong shocks
Ideal GLM-MHD - a new mathematical model for simulating astrophysical plasmas
Magnetic fields are ubiquitous in space. As there is strong evidence that magnetic fields play an important role in a variety of astrophysical processes, they should not be neglected recklessly. However, analytic models in astrophysical either do often not take magnetic fields into account or can do this after limiting simplifications reducing their overall predictive power. Therefore, computational astrophysics has evolved as a modern field of research using sophisticated computer simulations to gain insight into physical processes.
The ideal MHD equations, which are the most often used basis for simulating magnetized plasmas, have two critical drawbacks: Firstly, they do not limit the growth of numerically caused magnetic monopoles, and, secondly, most numerical schemes built from the ideal MHD equations are not conformable with thermodynamics.
In my work, at the interplay of math and physics, I developed and presented the first thermodynamically consistent model with effective inbuilt divergence cleaning. My new Galilean-invariant model is suitable for simulating magnetized plasmas under extreme conditions as those typically encountered in astrophysical scenarios. The new model is called the "ideal GLM-MHD" equations and supports nine wave solutions.
The accuracy and robustness of my numerical implementation are demonstrated with a number of tests, including comparisons to other schemes available within in the multi-physics, multi-scale adaptive mesh refinement (AMR) simulation code FLASH. A possible astrophysical application scenario is discussed in detail
FISH: A 3D parallel MHD code for astrophysical applications
FISH is a fast and simple ideal magneto-hydrodynamics code that scales to ~10
000 processes for a Cartesian computational domain of ~1000^3 cells. The
simplicity of FISH has been achieved by the rigorous application of the
operator splitting technique, while second order accuracy is maintained by the
symmetric ordering of the operators. Between directional sweeps, the
three-dimensional data is rotated in memory so that the sweep is always
performed in a cache-efficient way along the direction of contiguous memory.
Hence, the code only requires a one-dimensional description of the conservation
equations to be solved. This approach also enable an elegant novel
parallelisation of the code that is based on persistent communications with MPI
for cubic domain decomposition on machines with distributed memory. This scheme
is then combined with an additional OpenMP parallelisation of different sweeps
that can take advantage of clusters of shared memory. We document the detailed
implementation of a second order TVD advection scheme based on flux
reconstruction. The magnetic fields are evolved by a constrained transport
scheme. We show that the subtraction of a simple estimate of the hydrostatic
gradient from the total gradients can significantly reduce the dissipation of
the advection scheme in simulations of gravitationally bound hydrostatic
objects. Through its simplicity and efficiency, FISH is as well-suited for
hydrodynamics classes as for large-scale astrophysical simulations on
high-performance computer clusters. In preparation for the release of a public
version, we demonstrate the performance of FISH in a suite of astrophysically
orientated test cases.Comment: 27 pages, 11 figure
Trinity: A Unified Treatment of Turbulence, Transport, and Heating in Magnetized Plasmas
To faithfully simulate ITER and other modern fusion devices, one must resolve
electron and ion fluctuation scales in a five-dimensional phase space and time.
Simultaneously, one must account for the interaction of this turbulence with
the slow evolution of the large-scale plasma profiles. Because of the enormous
range of scales involved and the high dimensionality of the problem, resolved
first-principles global simulations are very challenging using conventional
(brute force) techniques. In this thesis, the problem of resolving turbulence
is addressed by developing velocity space resolution diagnostics and an
adaptive collisionality that allow for the confident simulation of velocity
space dynamics using the approximate minimal necessary dissipation. With regard
to the wide range of scales, a new approach has been developed in which
turbulence calculations from multiple gyrokinetic flux tube simulations are
coupled together using transport equations to obtain self-consistent,
steady-state background profiles and corresponding turbulent fluxes and
heating. This approach is embodied in a new code, Trinity, which is capable of
evolving equilibrium profiles for multiple species, including electromagnetic
effects and realistic magnetic geometry, at a fraction of the cost of
conventional global simulations. Furthermore, an advanced model physical
collision operator for gyrokinetics has been derived and implemented, allowing
for the study of collisional turbulent heating, which has not been extensively
studied. To demonstrate the utility of the coupled flux tube approach,
preliminary results from Trinity simulations of the core of an ITER plasma are
presented.Comment: 187 pages, 53 figures, Ph.D. thesis in physics at University of
Maryland, single-space versio
- …