Eric Sonnendrücker

Daniel Bach, Andrés M. Rueda-Ramírez, Eric Sonnendrücker et al.

We demonstrate that we can carry over the strategy of Finite Element Exterior Calculus (FEEC) to Summation-by-Parts (SBP) Finite Difference (FD) methods to achieve divergence- and curl-free discretizations. This is not obvious at first sight, as for SBP-FD no basis functions are known, but only values and derivatives at points. The key is a remarkable analytic relationship that enables us to construct compatible operators using integral and nodal degrees of freedom. Pre-existing SBP-FD matrix operators can then be used to obtain nodal values from the integral degrees of freedom to derive a scheme with the desired properties.

Nicolas Crouseilles, Thomas Respaud, Eric Sonnendrücker

This work deals with the numerical solution of the Vlasov equation. This equation gives a kinetic description of the evolution of a plasma, and is coupled with Poisson's equation for the computation of the self-consistent electric field. The coupled model is non linear. A new semi-Lagrangian method, based on forward integration of the characteristics, is developed. The distribution function is updated on an eulerian grid, and the pseudo-particles located on the mesh's nodes follow the characteristics of the equation forward for one time step, and are deposited on the 16 nearest nodes. This is an explicit way of solving the Vlasov equation on a grid of the phase space, which makes it easier to develop high order time schemes than the backward method.

Guillaume Latu, Virginie Grandgirard, Jérémie Abiteboul et al.

Inaccurate description of the equilibrium can yield to spurious effects in gyrokinetic turbulence simulations. Also, the Vlasov solver and time integration schemes impact the conservation of physical quantities, especially in long-term simulations. Equilibrium and Vlasov solver have to be tuned in order to preserve constant states (equilibrium) and to provide good conservation property along time (mass to begin with). Several illustrative simple test cases are given to show typical spurious effects that one can observes for poor settings. We explain why Forward Semi-Lagrangian scheme bring us some benefits. Some toroidal and cylindrical GYSELA runs are shown that use FSL.

Bedros Afeyan, Fernando Casas, Nicolas Crouseilles et al.

KEEN waves are nonlinear, non-stationary, self-organized asymptotic states in Vlasov plasmas outside the scope or purview of linear theory constructs such as electron plasma waves or ion acoustic waves. Nonlinear stationary mode theories such as those leading to BGK modes also do not apply. The range in velocity that is strongly perturbed by KEEN waves depends on the amplitude and duration of the ponderomotive force used to drive them. Smaller amplitude drives create highly localized structures attempting to coalesce into KEEN waves. These cases have much more chaotic and intricate time histories than strongly driven ones. The narrow range in which one must maintain adequate velocity resolution in the weakly driven cases challenges xed grid numerical schemes. What is missing there is the capability of resolving locally in velocity while maintaining a coarse grid outside the highly perturbed region of phase space. We here report on a new Semi-Lagrangian Vlasov-Poisson solver based on conservative non-uniform cubic splines in velocity that tackles this problem head on. An additional feature of our approach is the use of a new high-order time-splitting scheme which allows much longer simulations per computational e ort. This is needed for low amplitude runs which take a long time to set up KEEN waves, if they are able to do so at all. The new code's performance is compared to uniform grid simulations and the advantages quanti ed. The birth pains associated with KEEN waves which are weakly driven is captured in these simulations. These techniques allow the e cient simulation of KEEN waves in multiple dimensions which will be tackled next as well as generalizations to Vlasov-Maxwell codes which are essential to understanding the impact of KEEN waves in practice.

Emmanuel Frénod, Francesco Salvarani, Eric Sonnendrücker

We study the two-scale asymptotics for a charged beam under the action of a rapidly oscillating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution.

Emmanuel Franck, Matthias Hoelzl, Alexander Lessig et al.

In this paper we present a rigorous derivation of the reduced MHD models with and without parallel velocity that are implemented in the non-linear MHD code JOREK. The model we obtain contains some terms that have been neglected in the implementation but might be relevant in the non-linear phase. These are necessary to guarantee exact conservation with respect to the full MHD energy. For the second part of this work, we have replaced the linearized time stepping of JOREK by a non-linear solver based on the Inexact Newton method including adaptive time stepping. We demonstrate that this approach is more robust especially with respect to numerical errors in the saturation phase of an instability and allows to use larger time steps in the non-linear phase.

Jerome Guterl, Jean-Philippe Braeunig, Nicolas Crouseilles et al.

The purpose of this work is simulation of magnetised plasmas in the ITER project framework. In this context, Vlasov-Poisson like models are used to simulate core turbulence in the tokamak in a toroidal geometry. This leads to heavy simulation because a 6D dimensional problem has to be solved, 3D in space and 3D in velocity. The model is reduced to a 5D gyrokinetic model, taking advantage of the particular motion of particles due to the presence of a strong magnetic field. However, accurate schemes, parallel algorithms need to be designed to bear these simulations. This paper describes a Hermite formulation of the conservative PSM scheme which is very generic and allows to implement different semi-Lagrangian schemes. We also test and propose numerical limiters which should improve the robustness of the simulations by diminishing spurious oscillations. We only consider here the 4D drift-kinetic model which is the backbone of the 5D gyrokinetic models and relevant to build a robust and accurate numerical method.

Nicolas Besse, Francis Filbet, Michael Gutnic et al.

In this paper, we present very first results for the adaptive solution on a grid of the phase space of the Vlasov equation arising in particles accelarator and plasma physics. The numerical algorithm is based on a semi-Lagrangian method while adaptivity is obtained using multiresolution analysis.

Emmanuel Frénod, Alexandre Mouton, Eric Sonnendrücker

Motivated by the difficulty to solve numerically the weakly compressible 1D isentropic Euler equations with classical methods, we develop in this paper a two scale numerical method on this model. This method is based on two scale convergence theory developped by N'Guetseng and Allaire, and finite volume scheme. Furthermore, we do some numerical simulations in order to verify that the two-scale numerical method is more and more accurate when the Mach number diminishes.

Michael Kraus, Katharina Kormann, Philip J. Morrison et al.

We present a novel framework for Finite Element Particle-in-Cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi identity, as well as conservation of its Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from Finite Element Exterior Calculus ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge invariant, feature exact charge conservation and show excellent long-time energy and momentum behaviour. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the Finite Element basis, as long as the corresponding Finite Element spaces satisfy certain compatibility conditions.

Thomas Respaud, Eric Sonnendrücker

The Vlasov equation is a kinetic model describing the evolution of charged particles, and is coupled with Poisson's equation, which rules the evolution of the self-consistent electric field. In this paper, we introduce a new class of forward Semi-Lagrangian schemes for the Vlasov-Poisson system based on a Cauchy Kovalevsky (CK) procedure for the numerical solution of the characteristic curves. Exact conservation properties of the first moments of the distribution function for the schemes are derived and a convergence study is performed that applies as well for the CK scheme as for a more classical Verlet scheme. The convergence in L1 norm of the schemes is proved and error estimates are obtained.