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Solving the system of radiation magnetohydrodynamics for solar physical simulations in 3d [Elektronische Ressource] / Andreas Dedner

305 pages
Albert-Ludwigs-Universit¨at Freiburg i. Br.Fakult¨at fur¨ Mathematik und PhysikSolving the System of RadiationMagnetohydrodynamicsfor solar physical simulations in 3dAndreas DednerDissertation zur Erlangung des Doktorgrades der Fakult¨at fur¨ Mathematik undPhysik der Albert-Ludwigs-Universitat¨ Freiburg im BreisgauBetreuer: Prof. Dr. Dietmar Kroner¨Abteilung fur¨ Angewandte MathematikFreiburg im Breisgau, April 2003Dekan : Prof. Dr. Rolf SchneiderReferenten : Prof. Dr. Dietmar Kr¨oner: Prof. Dr. Gerald Warnecke, Universit¨at MagdeburgDatum der Promotion : 22. September 2003Picture on title page:simulation of a circular slip stream. In the purely hydrodynamic setting the interface is unstable withrespect to perturbations (Kelvin–Helmholtz instability). By means of a sufficiently strong magneticfield that is tangential to the interface, this instability can be suppressed. For the simulation shownhere, the magnetic field is not yet strong enough so that the development of the Kelvin–Helmholtzinstabilities can be clearly seen. The large picture shows the density of the fluid. The two smallerpictures show the locally adapted grid (top) and the third component of the magnetic field (bottom)in a small section of the domain in the vicinity of the interface.AbstractIn this study we present a finite–volume scheme for solving the equa-tions of radiation magnetohydrodynamics in two and three space di-mensions.
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Albert-Ludwigs-Universit¨at Freiburg i. Br.
Fakult¨at fur¨ Mathematik und Physik
Solving the System of Radiation
for solar physical simulations in 3d
Andreas Dedner
Dissertation zur Erlangung des Doktorgrades der Fakult¨at fur¨ Mathematik und
Physik der Albert-Ludwigs-Universitat¨ Freiburg im Breisgau
Betreuer: Prof. Dr. Dietmar Kroner¨
Abteilung fur¨ Angewandte Mathematik
Freiburg im Breisgau, April 2003Dekan : Prof. Dr. Rolf Schneider
Referenten : Prof. Dr. Dietmar Kr¨oner
: Prof. Dr. Gerald Warnecke, Universit¨at Magdeburg
Datum der Promotion : 22. September 2003
Picture on title page:
simulation of a circular slip stream. In the purely hydrodynamic setting the interface is unstable with
respect to perturbations (Kelvin–Helmholtz instability). By means of a sufficiently strong magnetic
field that is tangential to the interface, this instability can be suppressed. For the simulation shown
here, the magnetic field is not yet strong enough so that the development of the Kelvin–Helmholtz
instabilities can be clearly seen. The large picture shows the density of the fluid. The two smaller
pictures show the locally adapted grid (top) and the third component of the magnetic field (bottom)
in a small section of the domain in the vicinity of the interface.Abstract
In this study we present a finite–volume scheme for solving the equa-
tions of radiation magnetohydrodynamics in two and three space di-
mensions. Among other applications this system is used to model the
plasmainthesolarconvectionzoneandinthesolarphotosphere. Itis
a non–linear system of balance laws derived from the Euler equations
of gas dynamics and the Maxwell equations; the energy transport
through radiation is also included in the model. The starting point of
our presentation is a standard explicit first and second order finite–
volume scheme on both structured and unstructured grids. We first
study the convergence of a finite–volume scheme applied to a scalar
model problem for the full system of radiation magnetohydrodynam-
ics. We then present modifications of the base scheme. These make
anarbitraryequationofstate; theyreduceerrorsduetoaviolationof
the divergence constraint on the magnetic field, and they lead to an
improved accuracy in the approximation of solution near an equilib-
rium state. These modifications significantly increase the robustness
in the solar atmosphere. For simulations in the solar photosphere, we
have to take the radiation intensity into account. A scheme for solv-
ing the radiation transport equation is a further focus of this study.
We present both analytical results and numerical tests, comparing
our scheme with some standard schemes found in the literature. We
conclude our presentation with a study of the parallelization strategy
for distributed memory computers that we use in our 3d code.iiIntroduction
Numerical simulations have become an important tool for studying many different
physical and technical problems. Ranging from the formation of galaxies to weather
forecasts to the design for parts of complex machinery, the applications are numerous.
On the one hand, numerical simulations serve as a tool for the verification of physical
theories deduced from observation; on the other hand, they play an important role
in reducing development cost in manufacturing. Although the range of applications
is extremely broad, the methods used for solving problems numerically have many
features in common. This is due to the fact that the physical models used have similar
properties. For example, fluid flow in the atmosphere of stars or in car engines can be
modeled by very similar systems of equations and can be simulated using very similar
numerical methods.
In this study we investigate numerical schemes that can be used to simulate the evo-
lution of a compressible fluid. The governing system of partial differential equations
is based on the Euler equations of gas dynamics. Over the last centuries, this system
has been the focus of both analytical and numerical studies. A large number of dif-
ferent schemes have been developed and tested using this system. One very successful
approach turned out to be the finite–volume framework, and many different schemes
basedonthisapproachhavebeenpresented. Thesamemethodshavealsobeenapplied
to different extensions of the basic system of Euler equations, including, for example,
reactive flow and magnetohydrodynamics. The latter will be the main focus of our
Solar physical applications
The material presented in the following is part of a project financed by the Deutsche
methods for studying fluid flow in the solar atmosphere. The development of many of
the methods is a direct consequence of the interaction between members of our group
here in Freiburg (Dietmar Kroner,¨ Christian Rohde, Matthias Wesenberg, and myself)
and solar physicists (Manfred Schuss¨ ler and Peter Vollm¨oller from the Max–Planck
Institute for Aeronomie in Kattlenburg–Lindau), whose ideas greatly influenced our
work. Many of the problems discussed here occur only if the methods are applied
not to academic test cases, but to realistic settings. Therefore, the discussions with
the solar physicists and their help in developing and testing the numerical methods
influenced the direction in which our work progressed.
Although a variation in solar activity has a strong impact on life here on earth, a
thorough understanding of the physical processes behind these phenomena is still the
subject of research all over the world:
http://image.gsfc.nasa.gov/poetry/storm0/black1.html http://sec.noaa.gov/SWN/index.html
Storms are usually responsible for the losses of Early records of sunspots indicate that the Sun
electricity we endure, but did you know that went through a period of inactivity in the late
”storms” as far away as the sun are capable 17thcentury. Veryfewsunspotswereseenonthe
of knocking out large areas of electric service? Sun from about 1645 to 1715. [...] This period
Amazingly, the sun is capable of not only dis- of solar inactivity also corresponds to a climatic
rupting electrical power, but also short wave period called the ”Little Ice Age” when rivers
radio, television and telegraph signals, naviga- that are normally ice-free froze and snow fields
tional equipment (GPS and LORAN), defense remained year-round at lower altitudes. There
(military) early warning radar systems, the cli- is evidence that the Sun has had similar peri-
mate,andcanevenknockoutourcommunication ods of inactivity in the more distant past. The
satellites in space. connection between solar activity and terrestrial
http://image.gsfc.nasa.gov/poetry/storm0/black1.html climate is an area of on-going research.
Since the possibilities for direct observation of physical processes below the solar sur-
face are limited, numerical simulations play an important role in obtaining a clearer
understanding of solar phenomena. A further example of a solar phenomena not yet
fully understood is the eleven year cycle in which the number of sun spots on the solar
surfaceincreaseanddecrease. Onedifficultyisthatthefilamentsattheboundaryofthe
5sun spots are made up of magnetic fluxtubes that are formed about 2·10 kilometers
below the solar surface in the lower convection zone of the sun. In this region direct
observation is hardly possible so that the formation and evolution of the fluxtubes has
to be studied by means of numerical simulations. Although the presentation here is far
more general and such solar phenomena are not the immediate focus, the application
of our method to problems in solar physics has been a constant motivation.
Mathematical model
magnetohydrodynamics (MHD) and the radiation transport (RT) equation. The MHD
equations are a non–linear system of eight conservation laws; the energy transport
through radiation leads to an additional source term that is non–local in space.
The MHD equations describe the evolution of an electrically conductive plasma in the
presence of magnetic fields and combine the Euler equations of gas dynamics and theINTRODUCTION v
Maxwellequations. Thelatteralsointroduceaconstraintequationonthedivergenceof
the magnetic field. In the solar atmosphere the force of gravity plays an important role
and is included in our model via source terms. To perform the simulations, we have to
prescribe suitable initial conditions for the fluid. These often consist of a perturbation
of a stratified and static background atmosphere. One intrinsic problem of simulations
with this type of initial data is the size of the computational domain. The setting
allows for no physical boundaries, and the construction of suitable artificial boundary
conditions that can be used in numerical simulations is no easy undertaking.
The main difficulty in approximating the radiation field is the high dimensionality
of the problem and the propagation speed of the radiation, which is several orders
of magnitude above the speed of the fluid. In our model we deal with the second
problem by assuming an instantaneous radiation equilibrium. We have thus removed
the different time scales, but we introduce a non–local dependency into our problem,
which we have to cope with in our numerical scheme. The high dimensionality of the
radiation intensity — it depends on space, time, propagation angle, and frequency —
forces us to construct a very efficient solver to compute the radiation field.
Numerical scheme
We use a first and second order finite–volume scheme on locally adapted structured
and unstructured grids. We have implemented this method in one, two, and three
space dimensions, using both Cartesian and triangular grids in 2d and hexahedral
and tetrahedral grids in 3d. To increase the efficiency of the scheme, we make use
of parallelization strategies including distributed memory parallelization with dynamic
loadbalancing. Mostofthemethodspresentedinthisstudyare,however,notrestricted
existing method for solving the system of magnetohydrodynamics. We have already
pointed out that the application to solar physical problems serves as a motivation for
the development and test of the scheme; the presentation, however, is kept at a far
moregenerallevel. Forexample, thecorrectionmethodusedtocomputesolutionsnear
different applications where the problem of balancing source terms and flux gradients
plays a crucial role.
An important consideration for the development of our methods is their simple im-
plementation within the framework of an existing numerical scheme, which we modify
as little as possible. The complexity of our applications also requires an efficient so-
lution algorithm. Consequently, none of the modifications should lead to an increase
in the computational cost. Furthermore, we try to reduce the number of free param-
eters as much as possible; if available, we use an analytically motivated choice for the
parameters, otherwise we try to find suitable values by means of numerical tests.
Analytical justification
There are very few analytical results for complex non–linear coupled systems of the
type studied here. Even for the MHD system without radiation very little is knownvi INTRODUCTION
concerning the existence and uniqueness of solutions for general initial data. Conse-
quently, convergence analysis for numerical schemes is not yet available. One approach
often used in the analysis of complex systems is to reduce the complexity (often down
to a scalar balance law), taking care to retain the important characteristic features of
the original system. A multitude of analytical results are available for scalar balance
laws, ranging from existence and uniqueness results to the convergence of numerical
schemes in higher space dimensions. We employ this approach to justify the use of a
finite–volume scheme to solve the coupled system of radiation magnetohydrodynam-
ics. We carefully derive a scalar balance law that includes a non–local operator. This
source term, which models the radiation transport, is the novel feature of our model
problem. We first study the influence of this non–local term on the solution of the
model problem; then we prove the convergence of a finite–volume scheme including an
explicit approximation of the non–local operator.
Numerical tests
Themathematicalmodelconsistsoftwoparts, onedescribingtheevolutionofthefluid
andtheothertheradiationfield. Theconstructionofournumericalschemesisbasedon
this splitting, which leadstoa MHDandaRT“module”. Thesemodulesare discussed
and tested separately since the coupling of the radiation and the fluid flow occurs only
on a source term level. We thereby assume that the performance of the full scheme
can be measured by the performance of both contributing modules. This indirect test
of our algorithm is necessary since we are not aware of any simple test cases for the
full coupled system. A rigorous test of the full algorithm is very difficult and requires
a detailed understanding of the underlying physical processes in the solar atmosphere;
this is beyond the scope of this presentation.
The main focus of our study is a comparison of the efficiency of different numerical
schemes. We compare often used approaches from the literature with newly developed
schemes. We measure the efficiency of a scheme by studying the error to runtime ratio.
Since the complexity of our problems (especially of our simulations in 3d) leads to a
high demand on computational cost, the runtime efficiency of the numerical scheme
has to be the essential aspect of our study.
Hardware and software used
The numerical scheme is implemented in C++. We used many different computer
systems for our numerical tests, including single processor Linux PC, a shared memory
SGI computer system (Origin with 46 processors), the IBM RS/6000 SP computer
at the Rechenzentrum in Karlsruhe, and the IBM Regatta at the Rechenzentrum in
Freiburg. BothGnuPlotandthegraphicslibraryGraPEwereusedforthevisualization
of the data. Detailed references to all software packages used are given later.INTRODUCTION vii
Outline of the thesis
In the first chapter we derive the relevant system of equations for our solar physical
applications. The physical derivation of the full system is not discussed in detail,
only the relevant notation is introduced. The main part of our study is divided into
three parts, each of which is preceded by an overview chapter and concluded with a
summary. In the first part we outline a very general numerical scheme for solving the
system of radiation magnetohydrodynamics. We justify the numerical scheme through
the analysis of a simplified setting. In the second and third parts, we extend this basic
numericalscheme, treatingthepartsforthefluidandtheradiationseparately. Wenow
describe the three parts in more detail.
In the first part (Chapters 2–5) we present a standard finite–volume scheme for
solving the system of radiation magnetohydrodynamics. At this stage we describe only
the standard building block as can be found in the literature. The scheme described
in Chapter 3 does not yet include all aspects of our mathematical model and in its
basic form it is not suitable for use in challenging applications. It serves rather as the
skeleton for the extensions described in the second and third parts. Before we study
the necessary modifications of the scheme, we justify the general approach with an
analyticalstudyofasimplifiedmodelproblem. Thefactthatthecentralnon–standard
aspect of the system is the non–local effect of the radiation source term dictates the
choice of material presented in Chapter 4. The model problem consists of a scalar
balance law with a right hand side including a non–local integral operator. We
first study the properties of special solutions to the model scalar balance law and then
present a general convergence proof for finite–volume schemes in 2d.
In the second part (Chapters 6–10) we present modifications of that part of the numer-
ical scheme in which the evolution of the fluid variables is computed. In the overview
also describe approaches found in the literature, approaches we then use as comparison
schemes for our own solution technique. The comparison methods are chosen in accor-
extension of existing scheme and no additional computational cost). Chapters 7–9 are
devoted to the description of the methods and numerical tests for three central chal-
lenges: we first study a relaxation approach that allows us to extend a solver for a
perfect gas to approximate the MHD equations with a general equation of state;
netic field is coupled with the evolution equation for the magnetic field. Based on this
approach we derive a number of different correction mechanisms for reducing errors
in the divergence of the magnetic field. Finally we study a modification of the base
scheme that facilitates the accurate approximation of solutions near an equilibrium
study of numerical schemes for solving the radiation transport equation. Again
we start with an overview in which we discuss the central aspects of this part of the
numerical scheme and present a standard solver found in the literature that we use as
a reference method. In Chapter 12 we derive a numerical scheme for approximating
the radiation intensity for a fixed propagation direction. This is the central buildingviii INTRODUCTION
block used for the approximation of the radiation source term that enters into the
balance law for the total energy. We present a convergence proof for our method
and, after presenting numerical tests for fixed propagation directions, we conclude
our investigation of the radiation transport module in Chapter 13 by studying the
approximation of the radiation source term itself.
In the last Chapter we then present some results using our3d MHD code, including
a simulation for a problem from solar physics.
The enclosed CD ROM contains a pdf version of this thesis and the sources of our
2d and 3d MHD code. Furthermore, we have included the web pages of our project.
produced during the project. The file (MHD.html) in the root directory of the CD
ROM gives specific details of the layout.
I would like to express my thanks to all my colleagues here at the IAM in Freiburg
— especially to Matthias Wesenberg and Christian Rohde for the discussions of the
methods,oftheresults,andoflife,theuniverse,andeverything. Thereareinfactmany
reasons for thanking Matthias Wesenberg with whom I have been working together
since the beginning of our studies. All my colleagues helped me by supplying such
important raw material as coffee and tea; special thanks, however, should go to the
different people who, over the last few years, invested a huge amount of their time to
keeping our computer system in working order.
Let me continue by saying thank you to my supervisor Dietmar Kr¨oner for suggesting
andplanningthisprojectandforthemanyfruitfuldiscussionsandsuggestions. Iwould
also like to thank him for introducing me to several other scientists who also influenced
theworkpresentedhere, butofwhomIcanmentiononlyafewhere. Ontheonehand,
these include our project partners from the Max–Planck Institute of Aeronomie, Man-
fred Schu¨ssler and Peter Vollm¨oller. Many of the problems studied here were brought
from solar physics; the development of the new radiation transport solver was one of
the results of this cooperation. On the other hand, I want to mention Ivan Sofronov
from the Keldysh Institute of Applied Mathematics RAS in Moscow, with whom we
developed transparent boundary conditions for our atmospheric flow problems, and
Claus–Dieter Munz from the Institut fur¨ Aerodynamik und Gasdynamik, Universit¨at
Stuttgart, who suggested extending his divergence cleaning technique derived for the
Maxwell equations to the MHD system. The analytical results for the finite–volume
scheme were proven in cooperation with Christian Rohde, large parts of the numerical
scheme were implemented together with Matthias Wesenberg, and the grid concept
goes back to Bernhard Schupp, whom I would also like to thank for saving me from
the tedious task of having to design the hierarchical grid concept.
My thanks for proofreading this thesis go to Christian Rohde and to my mother. To
my wife Sabine Voigt go my very special thanks for reasons of which she is well aware.
This study was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of
thepriorityresearchprogramAnalysisandNumericsforConservationLaws (ANumE).