Vlasov-Poisson in Six Dimensions – VLASIX

The numerical methods we develop can be mainly classified in four categories:

- Pure Eulerian mesh based methods, where the distribution function is followed at fixed points of phase-space. Our approach combines upwind schemes with adaptive mesh refinement.

- Semi-Lagrangian methods. The phase-space distribution function is still sampled on a mesh, but test particles following motion are used to update it according to Liouville theorem. Our approaches rely on standard spline interpolation or discontinuous Galerkin decomposition with the possibility of adaptive mesh refinement to estimate the phase-space distribution function.

- Pure Lagrangian methods, that is following the flow in phase-space, using contour (waterbags) and adaptive tessellation dynamics.

- Hybrid methods, which combine for instance waterbags with semi-Lagrangian schemes.

In addition to developing codes, we study mathematical properties and convergence of various numerical schemes, compare Vlasov codes to the traditional N-body approach and apply our codes to specific problems where analytical theories can be tested. One of the goals of this project is for instance to confirm and to understand universal properties of dark matter halos.

The cornerstone of VLASIX project is the massively parallel public code ColDICE of which the purpose is to solve Vlasov-Poisson equations in the cold cosmological case. ColDICE samples the phase-space sheet with an adaptive simplicial tessellation (tetrahedra) following the motion. In addition to ColDICE, other products of VLASIX project are numerous, both from the numerical and the theoretical points of view, and we list the main ones here:

Codes and methods:

- A prototype of 6D code with adaptive mesh refinement using a high-order adaptive convection scheme.
- VlaSolve: a public semi-Lagrangian code in spherical symmetry.
- Vlamet: a new semi-Lagrangian scheme using metric elements to reduce diffusion.
- GalWa: a public 2D semi-Lagrangian code using adaptive mesh refinement with multiwavelets and discontinuous Galerkin representation.
- Vlapoly: a 2D Vlasov-Poisson solver using the waterbag method.
- Participation to the development of semi-lagrangian and discontinuous-Galerkin parallel codes for reduced models of waterbag-type (gyrokinetic-waterbag, waterbag-{Poisson, Maxwell, quasineutral}).

Theory and analysis:

- Comparisons Vlasov versus N-body.
- Mathematical analysis of kinetic models (e.g. Vlasov-Dirac-Benney, relativistic Vlasov-Maxwell), of reduced gyrokinetic models (gyrokinetic-waterbag) and fluid models (incompressible Euler, magnetohydrodynamics)
- Approximation and numerical simulations of gyrokinetic-waterbag models to study ITG instability and induced turbulence in fusion plasmas.
- Perturbation theory for the pre- and post-collapse dynamics of Vlasov-Poisson equations in cosmology, for incompressible Euler equations in a three-dimensional bounded domain and gyrokinetic-waterbag models.

Vlasov-Poisson in the cold case (dark matter):

The adaptive tessellation method used in ColDICE induces an augmentation of the number of sampling elements with time, which makes the code too costly to follow dark matter dynamics over a large number of dynamical times. An improvement of the code to make it workable in the general case consists in replacing simplices with particles when the fluid representation is not needed anymore, which is the case when the system is sufficiently evolved.

Vlasov in the warm case:

We could combine our adaptive discontinuous Galerkin semi-Lagrangian schemes with the metric approach to reduce even more diffusion and extend these schemes to the case of 4D and 6D Vlasov-Poisson, as well as other Vlasov type models, such as gyrokinetic-Vlasov and Vlasov-Maxwell (relativistic, or not) models.

Semi-Lagrangian methods applied to Euler equations:

We could extend Cauchy-Lagrange techniques to the case of incompressible Euler equations on a Riemannian manifold of arbitrary dimension (curved spaces) as well as models of magnetohydrodynamics in a flat or curved space. These techniques will allow us to study the regularity of the Lagrangian flow associated to these equations and conceive forward semi-Lagrangian methods of high order to solve them numerically.

Publications of the VLASIX project:

[1] Alard C., 2014, ApJ 788, 171
[2] Bardos C., Besse N., 2013, Kin. and Relat. Models 6, 893
[3] Bardos C., Besse N., 2016, Bull. Inst. Math. Acad. Sin. 11, 43
[4] Besse N., 2016, Commun. Math. Sci. 14, 593.
[5] Besse N., Coulette D., 2016, J. Math. Phys. 57, 081518
[6] Besse N., 2017, IMA J. Numer. Anal. 37, 985
[7] Besse N., Frisch U., 2017, J. Fluid. Mech. 825, 412
[8] Besse N., Frisch U., 2017, Commun. Math. Phys. 351, 689
[9] Besse N., Deriaz E., Madaule E., 2017, J. Comput. Phys 332, 376 (GalWa)
[10] Colombi S., Touma J., 2014, MNRAS 441, 2414 (Vlapoly)
[11] Colombi S., 2015, MNRAS 446, 2902
[12] Colombi S., Sousbie T., Peirani S., Plum G., Suto Y., 2015, MNRAS 450, 3724 (VlaSolve)
[13] Colombi S., Alard C., 2017, Journal of Plasma Physics 83, 705830302 (Vlamet)
[14] Coulette D., Besse N., 2017, J. Plasma Phys. 83, 905830207
[15] Deriaz E., Peirani S., 2018, accepted in Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal (6D code prototype)
[16] Sousbie T., Colombi S., 2016, Journal of Computational Physics 321, 644 (ColDICE)
[17] Taruya A., Colombi S., 2017, MNRAS 470, 4858

Submitted articles:
- Hallé A., Colombi S., Peirani S., eprint arXiv:1701.01384
- On regularity of weak solutions of the relativistic Vlasov-Maxwell system, Besse N., Bechouche P.

In our current view of cosmological structure formation, luminous objects form in gravitational potential wells of dark matter halos, of which the dynamics is governed by Vlasov-Poisson equations. The knowledge of the fine structure of dark matter halos is therefore essential to interpret results of astrophysical observational experiments, that include the measurement of rotation curves of spiral galaxies, weak and strong lensing, properties of galaxy clusters and direct/indirect detection of dark matter itself. Usually, dark matter dynamics is numerically modeled with a N-body approach, which is not free from significant defects due to the discrete representation of the phase-space distribution function with particles.

The main objective of this proposal is, instead of relying on a particle based approach, to develop and to use a new cosmological semi-Lagrangian direct Vlasov solver to study the formation and evolution of cold dark matter halos and to compare the results to N-body simulations. One goal of this project is to confirm universal properties of dark matter halos measured with N-body techniques and to understand them.

We shall proceed in two stages: in a first step, we shall restrict to spherical symmetry. In this framework, that reduces phase-space to 3 dimensions, we shall be able to test various semi-Lagrangian schemes to prepare ourselves for the full 6D challenge. With the spherical code, we shall conduct two specific projects: we shall try to understand how quasi-stationary states build up as functions of initial conditions and we shall investigate effects of N-body relaxation and force softening on the dynamics. In a second step, we shall develop a fully 6-dimensional massively parallel semi-Lagrangian Vlasov code. We shall compare the results obtained with that code to N-body results, both at large scales, paying attention to mass function of halos and power-spectrum of density fluctuations, and at small scales, focusing on the fine internal structure of dark matter halos.