Periodic Reporting for period 1 - SuPerMan (Structure Preserving schemes for Conservation Laws on Space Time Manifolds)
Okres sprawozdawczy: 2021-06-01 do 2023-05-31
In this research project our main objective concerns the development and the efficient implementation of new structure preserving high order numerical methods to solve complex systems of hyperbolic partial differential equations.
A special interest is devoted to systems written in covariant form, i.e. independent of the chosen coordinate system, describing phenomena on general spacetime manifolds. This is the case for some reformulations of simple Newtonian systems (such as Euler and Shallow Water equations) used in this project to benchmark our schemes. In addition, this is in particular the case for the systems of equations which describe astrophysical events as i) the General Relativistic Magneto-Hydrodynamics (GRMHD) system which describes the matter dynamics of for example neutron stars and accretion disks, according to a general metric; and this metric can be fixed or we can consider its evolution described by the ii) Einstein field equations of general relativity.
Being able to numerically simulate these complex systems could reveal novel information about the origin and the future of our Universe, and also increase our ability in modelling valuable events on Earth.
However, these systems are particularly challenging because i) they contain many complex unknowns and ii) their stability could be easily compromised by the accumulation of numerical errors. For these reasons the use of high order of accuracy or fine meshes alone, as done in standard codes, represents an unpracticable solution. Thus SuPerMan wants to incorporate in these codes additional structure preserving capabilities, that, in addition to accurately solving the equations, preserve exactly, at the discrete level, some physical and geometrical invariants of the studied continuum model.
A special interest is devoted to systems written in covariant form, i.e. independent of the chosen coordinate system, describing phenomena on general spacetime manifolds. This is the case for some reformulations of simple Newtonian systems (such as Euler and Shallow Water equations) used in this project to benchmark our schemes. In addition, this is in particular the case for the systems of equations which describe astrophysical events as i) the General Relativistic Magneto-Hydrodynamics (GRMHD) system which describes the matter dynamics of for example neutron stars and accretion disks, according to a general metric; and this metric can be fixed or we can consider its evolution described by the ii) Einstein field equations of general relativity.
Being able to numerically simulate these complex systems could reveal novel information about the origin and the future of our Universe, and also increase our ability in modelling valuable events on Earth.
However, these systems are particularly challenging because i) they contain many complex unknowns and ii) their stability could be easily compromised by the accumulation of numerical errors. For these reasons the use of high order of accuracy or fine meshes alone, as done in standard codes, represents an unpracticable solution. Thus SuPerMan wants to incorporate in these codes additional structure preserving capabilities, that, in addition to accurately solving the equations, preserve exactly, at the discrete level, some physical and geometrical invariants of the studied continuum model.
To pursue our objectives, we work in three directions.
The first, and let’s say classical, consists in investigating how to improve our high order numerical methods: during SuPerMan this led to 3 publications that account for novel boundary conditions (https://arxiv.org/abs/2209.14892(odnośnik otworzy się w nowym oknie)) novel basis functions (https://arxiv.org/abs/2205.14673(odnośnik otworzy się w nowym oknie)) and improving our a posteriori limiter (https://arxiv.org/abs/2010.04853(odnośnik otworzy się w nowym oknie)).
Then, particular attention has been devoted to improving our direct Arbitrary-Lagrangian-Eulerian (ALE) algorithm. Lagrangian algorithms allow to reduce the numerical dissipation at contact waves and moving material interfaces and guarantee the Galilean and rotational invariance, so they have very desirable structure preserving features. However, in these schemes the mesh moves together with the fluid flow, leading frequently to distortions that may slow down or even destroy the computation. Thus, to exploit the power of Lagrangian methods and always maintaining a high quality of the moving mesh, we have developed a ground-breaking novel approach that permits the use of mesh optimization techniques and to integrate with high order of accuracy when, in order to optimize the mesh, we introduce a so-called topology change. This approach, which makes use of integration on hole-like degenerate elements, is a novelty introduced by the ER in 2020 and further developed over the years. Further details can be found in https://arxiv.org/abs/2208.02092(odnośnik otworzy się w nowym oknie).
The third main activity of this research project concerns the introduction of the above described schemes of novel structure preserving techniques guaranteeing entropy stability (https://arxiv.org/abs/2206.03889(odnośnik otworzy się w nowym oknie)) and well-balancing. Well-balancing is a technique able to guarantee the exact preservation of equilibria thus allowing to model with higher accuracy the small physical perturbations happening around equilibria profiles.
We employed these techniques for different applications of increasing difficulties, many of which are unfeasible without the present technologies, thus representing a major enhancement in the community of computational astrophysics:
- For modelling Shallow Water equations in covariant coordinate (https://arxiv.org/abs/2209.01036)(odnośnik otworzy się w nowym oknie);
- For modelling GRMHD and Einstein field equations in 1D (https://arxiv.org/abs/2108.02960)(odnośnik otworzy się w nowym oknie);
- For increasing the capabilities of our direct ALE method with topology changes and modelling instabilities over Keplerian disks;
- For modelling GRMHD and Einstein field equations in 3D, being able to obtain remarkable results as the simulation of i) black holes with extreme spin, ii) TOV star evolved in pure vacuum and iii) head-on collision of two punctures black holes (https://arxiv.org/abs/2307.06629(odnośnik otworzy się w nowym oknie)).
(Note: for the last two sets of results, the complete publications will be available soon on the project website https://www.elenagaburro.it/SuPerMan.html(odnośnik otworzy się w nowym oknie) and on the preprint server ArXiv https://arxiv.org/(odnośnik otworzy się w nowym oknie)).
Next, all along the project, the ER Elena Gaburro, has disseminated her knowledge and the project results in 15 international conferences and 5 laboratory seminars and she has organized 1 international conference (https://www.math.uzh.ch/multimat2022(odnośnik otworzy się w nowym oknie)) one regional workshop (https://indico.math.cnrs.fr/event/7007/(odnośnik otworzy się w nowym oknie)) and 2 PhD schools.
The first, and let’s say classical, consists in investigating how to improve our high order numerical methods: during SuPerMan this led to 3 publications that account for novel boundary conditions (https://arxiv.org/abs/2209.14892(odnośnik otworzy się w nowym oknie)) novel basis functions (https://arxiv.org/abs/2205.14673(odnośnik otworzy się w nowym oknie)) and improving our a posteriori limiter (https://arxiv.org/abs/2010.04853(odnośnik otworzy się w nowym oknie)).
Then, particular attention has been devoted to improving our direct Arbitrary-Lagrangian-Eulerian (ALE) algorithm. Lagrangian algorithms allow to reduce the numerical dissipation at contact waves and moving material interfaces and guarantee the Galilean and rotational invariance, so they have very desirable structure preserving features. However, in these schemes the mesh moves together with the fluid flow, leading frequently to distortions that may slow down or even destroy the computation. Thus, to exploit the power of Lagrangian methods and always maintaining a high quality of the moving mesh, we have developed a ground-breaking novel approach that permits the use of mesh optimization techniques and to integrate with high order of accuracy when, in order to optimize the mesh, we introduce a so-called topology change. This approach, which makes use of integration on hole-like degenerate elements, is a novelty introduced by the ER in 2020 and further developed over the years. Further details can be found in https://arxiv.org/abs/2208.02092(odnośnik otworzy się w nowym oknie).
The third main activity of this research project concerns the introduction of the above described schemes of novel structure preserving techniques guaranteeing entropy stability (https://arxiv.org/abs/2206.03889(odnośnik otworzy się w nowym oknie)) and well-balancing. Well-balancing is a technique able to guarantee the exact preservation of equilibria thus allowing to model with higher accuracy the small physical perturbations happening around equilibria profiles.
We employed these techniques for different applications of increasing difficulties, many of which are unfeasible without the present technologies, thus representing a major enhancement in the community of computational astrophysics:
- For modelling Shallow Water equations in covariant coordinate (https://arxiv.org/abs/2209.01036)(odnośnik otworzy się w nowym oknie);
- For modelling GRMHD and Einstein field equations in 1D (https://arxiv.org/abs/2108.02960)(odnośnik otworzy się w nowym oknie);
- For increasing the capabilities of our direct ALE method with topology changes and modelling instabilities over Keplerian disks;
- For modelling GRMHD and Einstein field equations in 3D, being able to obtain remarkable results as the simulation of i) black holes with extreme spin, ii) TOV star evolved in pure vacuum and iii) head-on collision of two punctures black holes (https://arxiv.org/abs/2307.06629(odnośnik otworzy się w nowym oknie)).
(Note: for the last two sets of results, the complete publications will be available soon on the project website https://www.elenagaburro.it/SuPerMan.html(odnośnik otworzy się w nowym oknie) and on the preprint server ArXiv https://arxiv.org/(odnośnik otworzy się w nowym oknie)).
Next, all along the project, the ER Elena Gaburro, has disseminated her knowledge and the project results in 15 international conferences and 5 laboratory seminars and she has organized 1 international conference (https://www.math.uzh.ch/multimat2022(odnośnik otworzy się w nowym oknie)) one regional workshop (https://indico.math.cnrs.fr/event/7007/(odnośnik otworzy się w nowym oknie)) and 2 PhD schools.
Among the many results achieved during the project, two, in particular, have pushed forward the state of the art.
First, the development of our ground-breaking Arbitrary-Lagrangian-Eulerian method that completely changes the paradigm of Lagrangian schemes by making it possible to achieve robustness for any complex fluid motion and high order of accuracy while guaranteeing a minimal dissipation, conservation and satisfaction of the GCL. During SuPerMan we have proven that in 2D this is a perfect solution that deserves the major and risky theoretical and practical effort that the ER will dedicate to its 3D extension in her future research.
Second, during SuPerMan we have been able to numerically simulate, for the first time in literature, the first order hyperbolic reformulation of the Einstein field equation of general relativity coupled with the GRMHD system. This opens new horizons for the simulation of astrophysical events making it possible to model the coupled evolution of matter and metric inside a unified framework by means of high order discontinuous Galerkin schemes with structure preserving properties.
First, the development of our ground-breaking Arbitrary-Lagrangian-Eulerian method that completely changes the paradigm of Lagrangian schemes by making it possible to achieve robustness for any complex fluid motion and high order of accuracy while guaranteeing a minimal dissipation, conservation and satisfaction of the GCL. During SuPerMan we have proven that in 2D this is a perfect solution that deserves the major and risky theoretical and practical effort that the ER will dedicate to its 3D extension in her future research.
Second, during SuPerMan we have been able to numerically simulate, for the first time in literature, the first order hyperbolic reformulation of the Einstein field equation of general relativity coupled with the GRMHD system. This opens new horizons for the simulation of astrophysical events making it possible to model the coupled evolution of matter and metric inside a unified framework by means of high order discontinuous Galerkin schemes with structure preserving properties.