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+
+
+
+ 20231211T163126-524309b25d53e003319613c3ba9b5cbfe680c7f0
+ 20231211163126
+
+ JOSS Admin
+ admin@theoj.org
+
+ The Open Journal
+
+
+
+
+ Journal of Open Source Software
+ JOSS
+ 2475-9066
+
+ 10.21105/joss
+ https://joss.theoj.org
+
+
+
+
+ 12
+ 2023
+
+
+ 8
+
+ 92
+
+
+
+ GECo: A collection of solvers for the self-gravitating
+Vlasov equations
+
+
+
+ Ellery
+ Ames
+ https://orcid.org/0000-0001-9444-585X
+
+
+ Anders
+ Logg
+ https://orcid.org/0000-0002-1547-4773
+
+
+
+ 12
+ 11
+ 2023
+
+
+ 5979
+
+
+ 10.21105/joss.05979
+
+
+ http://creativecommons.org/licenses/by/4.0/
+ http://creativecommons.org/licenses/by/4.0/
+ http://creativecommons.org/licenses/by/4.0/
+
+
+
+ Software archive
+ 10.5281/zenodo.10351519
+
+
+ GitHub review issue
+ https://github.com/openjournals/joss-reviews/issues/5979
+
+
+
+ 10.21105/joss.05979
+ https://joss.theoj.org/papers/10.21105/joss.05979
+
+
+ https://joss.theoj.org/papers/10.21105/joss.05979.pdf
+
+
+
+
+
+ Automated Solution of Differential Equations
+by the Finite Element Method
+ 84
+ 10.1007/978-3-642-23099-8
+ 978-3-642-23099-8
+ 2012
+ Logg, A., Mardal, K.-A., & Wells,
+G. (Eds.). (2012). Automated Solution of Differential Equations by the
+Finite Element Method (Vol. 84). Springer Berlin Heidelberg.
+https://doi.org/10.1007/978-3-642-23099-8
+
+
+ On axisymmetric and stationary solutions of
+the self-gravitating Vlasov system
+ Ames
+ Class. Quantum Grav.
+ 15
+ 33
+ 10.1088/0264-9381/33/15/155008
+ 2016
+ Ames, E., Andréasson, H., & Logg,
+A. (2016). On axisymmetric and stationary solutions of the
+self-gravitating Vlasov system. Class. Quantum Grav., 33(15), 155008.
+https://doi.org/10.1088/0264-9381/33/15/155008
+
+
+ The Einstein-Vlasov System/Kinetic
+Theory
+ Andréasson
+ Living Reviews in Relativity
+ 14
+ 10.12942/lrr-2011-4
+ 2011
+ Andréasson, H. (2011). The
+Einstein-Vlasov System/Kinetic Theory. Living Reviews in Relativity, 14.
+https://doi.org/10.12942/lrr-2011-4
+
+
+ Cosmic string and black hole limits of
+toroidal Vlasov bodies in general relativity
+ Ames
+ Physical review D
+ 2
+ 99
+ 10.1103/PhysRevD.77.124044
+ 2019
+ Ames, E., Andréasson, H., & Logg,
+A. (2019). Cosmic string and black hole limits of toroidal Vlasov bodies
+in general relativity. Physical Review D, 99(2), 024012.
+https://doi.org/10.1103/PhysRevD.77.124044
+
+
+ Galactic Dynamics: Second
+Edition
+ Binney
+ 0691130264
+ 2008
+ Binney, J., & Tremaine, S.
+(2008). Galactic Dynamics: Second Edition. Princeton University Press.
+ISBN: 0691130264
+
+
+ Anderson acceleration for fixed-point
+iterations
+ Walker
+ SIAM Journal of Numerical
+Analysis
+ 49
+ 10.1137/10078356X
+ 2011
+ Walker, H. F., & Ni, P. (2011).
+Anderson acceleration for fixed-point iterations. SIAM Journal of
+Numerical Analysis, 49, 1715.
+https://doi.org/10.1137/10078356X
+
+
+
+
+
+
diff --git a/joss.05979/10.21105.joss.05979.jats b/joss.05979/10.21105.joss.05979.jats
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+
+
+
+
+
+
+
+Journal of Open Source Software
+JOSS
+
+2475-9066
+
+Open Journals
+
+
+
+5979
+10.21105/joss.05979
+
+GECo: A collection of solvers for the self-gravitating
+Vlasov equations
+
+
+
+https://orcid.org/0000-0001-9444-585X
+
+Ames
+Ellery
+
+
+
+
+https://orcid.org/0000-0002-1547-4773
+
+Logg
+Anders
+
+
+
+
+
+Flax and Teal
+
+
+
+
+Chalmers University of Technology
+
+
+
+
+20
+9
+2023
+
+8
+92
+5979
+
+Authors of papers retain copyright and release the
+work under a Creative Commons Attribution 4.0 International License (CC
+BY 4.0)
+2022
+The article authors
+
+Authors of papers retain copyright and release the work under
+a Creative Commons Attribution 4.0 International License (CC BY
+4.0)
+
+
+
+Einstein-Vlasov
+
+
+
+
+
+ Summary
+
Gothenburgh Einstein solver Collection (GECo) is a collection of
+ solvers for stationary self-gravitating collisionless kinetic (Vlasov)
+ matter. The gravitational interaction may be taken to be either
+ Newtonian or general relativistic. GECo is focused on the solutions
+ which are axisymmetric, meaning that the gravitational and matter
+ fields have a rotational symmetry. In this setting stationary
+ solutions may be generated with the choice of a particular ansatz
+ function for the Vlasov distribution function. GECo allows users to
+ easily introduce new ansatz functions and explore the properties of
+ the resulting stationary solutions.
+
+
+ Statement of need
+
In understanding a physical model one usually starts with a
+ simplified setting, such as by imposing symmetry assumptions. In the
+ case of self-gravitating kinetic matter, stationary solutions in the
+ spherically symmetric setting are well understood
+ (Andréasson,
+ 2011;
+ Binney
+ & Tremaine, 2008). However, many of the physical systems of
+ interest such as accretion disks, galaxies, galaxy clusters and so on,
+ require models beyond spherical symmetry. When going beyond spherical
+ symmetry, the coupled and nonlinear PDE systems in high dimensions –
+ such as the self-gravitating Vlasov equations – are difficult to
+ investigate analytically, and numerical approaches are essential to
+ understand behavior of solutions and to answer questions of physical
+ and mathematical interest. The GECo code started with the desire to
+ understand properties of stationary and axisymmetric solutions of the
+ Einstein-Vlasov system.
+
+
+ Method and implementation
+
To construct stationary solutions, the code relies on a reduction
+ method in which the distribution function for the matter is assumed to
+ depend on the position and momentum phase-space coordinates solely
+ through conserved quantities, such as the particle energy and angular
+ momentum about the axis of symmetry. With this ansatz the
+ Einstein–Vlasov or Vlasov–Poisson system (depending on the
+ gravitational model used) forms a semi-linear integro-differential
+ system of equations. In GECo, the form of the ansatz is called a
+ MaterialModel and several different choices are
+ implemented as subclasses of the FEniCS/DOLFIN Expression class. The
+ semi-linear integro-differential system is solved via a
+ mass-preserving fixed point scheme using Anderson acceleration
+ (Walker
+ & Ni, 2011). At each step of the fixed point method, the
+ linear system of equations is solved using finite elements implemented
+ with the FEniCS toolkit
+ (Logg
+ et al., 2012). The computational domain is taken to be the
+ half-meridional plane
+ 0, z>0 \}]]>
+ {(r,z):r>0,z>0}
+ in cylindrical coordinates, with a semi-circular outer boundary; see
+ [fig:Solution].
+ Details of the mathematical formulation and implementation can be
+ found in
+ (Ames
+ et al., 2016)
+
+
+ Functionality
+
The entrypoint for GECo is a run script written in Python. In this
+ file, the user selects the solver class
+ (EinsteinVlasovSolver or
+ VlasovPoisson) that specifies the model for the
+ gravitational interaction, a MaterialModel to
+ specify the particular form of the reduction ansatz, and several
+ parameters related to the model and discretization. Calling the
+ solve method within the script invokes the
+ solver to construct a stationary solution via the fixed point scheme
+ mentioned above, which runs until convergence within a specified
+ tolerance. Gravitational fields and matter quantities are saved in
+ XMDF and XML format that can be consumed by visualization software
+ like Paraview and VisIT, as well as postprocessing scripts.
+ Multi-component solutions may be constructed from multiple
+ MaterialModels by combining models in a
+ weighted sum.
+
GECo includes several postprocessing routines that:
+
+
+
generate additional scalar data not computed during the fixed
+ point iteration;
+
+
+
represent the matter density as well as an ergoregion (if
+ present) in
+
+ ℝ2
+ (i.e. reflected about the reflection plane and symmetry axis), as
+ shown in
+ [fig:2Ddensity];
+
+
+
represent the matter density as well as an ergoregion (if
+ present) as a volume in
+
+ ℝ3,
+ facilitating visualization of contours, as shown in
+ [fig:3Ddensity];
+
+
+
represent the density as a three-dimensional point cloud, as
+ shown in
+ [fig:PointCloud];
+
+
+
compute the Kretschmann curvature scalar.
+
+
+
+
Computed spatial density of torus solution on the
+ quarter plane computational domain.
+
+
+
+
+
Computed spatial density of torus solution extended to
+
+
+ xy-plane.
+
+
+
+
+
Computed spatial density of torus solution visualized as
+ iso surfaces in 3D.
+
+
+
+
+
Computed spatial density of torus solution visualized as
+ a point cloud.
+
+
+
+
+
+ Documentation
+
The documentation for GECo is published on the
+ GECo
+ GitHub pages.
+
+
+ Limitations and future work
+
We briefly list a few directions of interest for future work.
+
+
+
GECo currently uses a uniform mesh. However, in axisymmetry
+ (unlike spherical symmetry) the solution is not uniquely defined
+ outside the support of the matter, and asymptotically flat
+ boundary conditions must be applied sufficiently far from the
+ matter. An adaptive mesh refinement algorithm was developed and
+ used in
+ (Ames
+ et al., 2019) to investigate properties of extreme rotating
+ toroidal solutions. It remains however to integrate such an
+ adaptive mesh refinement scheme into the core of GECo.
+
+
+
Currently the particles only interact via the gravitational
+ field generated by the particle distribution. An exciting area at
+ the frontier of astrophysics currently is the study of accretion
+ disks, where both central black holes and electromagnetic fields
+ play important roles. To lay groundwork for this area in
+ fundamental relativity, it is thus highly desirable to extend GECo
+ to the Einstein-Vlasov-Maxwell system and allow the inclusion of
+ central black holes.
+
+
+
While multi-species solutions can be generated in which the
+ different species follow different distribution ansatzes, the
+ particle properties are otherwise taken to be the same.
+ Astrophysical systems however often consist of particle-like
+ entities with very different properties (such as stars and dust).
+ We thus propose to allow different particle species to have
+ different particle properties such as mass and charge.
+
+
+
+
+
+
+
+
+ Automated Solution of Differential Equations by the Finite Element Method
+
+ LoggAnders
+ MardalKent-Andre
+ WellsGarth
+
+ Springer Berlin Heidelberg
+ Berlin, Heidelberg
+ 2012
+ 84
+ 978-3-642-23099-8
+ 10.1007/978-3-642-23099-8
+
+
+
+
+
+ AmesEllery
+ AndréassonHåkan
+ LoggAnders
+
+ On axisymmetric and stationary solutions of the self-gravitating Vlasov system
+ Class. Quantum Grav.
+ IOP Publishing
+ 201607
+ 33
+ 15
+ 10.1088/0264-9381/33/15/155008
+ 155008
+
+
+
+
+
+
+ AndréassonHåkan
+
+ The Einstein-Vlasov System/Kinetic Theory
+ Living Reviews in Relativity
+ 2011
+ 14
+ 10.12942/lrr-2011-4
+
+
+
+
+
+ AmesEllery
+ AndréassonHåkan
+ LoggAnders
+
+ Cosmic string and black hole limits of toroidal Vlasov bodies in general relativity
+ Physical review D
+ 201901
+ 99
+ 2
+ 10.1103/PhysRevD.77.124044
+ 024012
+
+
+
+
+
+
+ BinneyJames
+ TremaineScott
+
+ Galactic Dynamics: Second Edition
+ Princeton University Press
+ 2008
+ 0691130264
+
+
+
+
+
+ WalkerHomer F.
+ NiPeng
+
+ Anderson acceleration for fixed-point iterations
+ SIAM Journal of Numerical Analysis
+ 2011
+ 49
+ 10.1137/10078356X
+ 1715
+
+
+
+
+
+
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