Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code

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

SPLINE DISCRETE DIFFERENTIAL FORMS AND A NEW FINITE DIFFERENCE DISCRETE HODGE OPERATOR

We construct a new set of discrete differential forms based on B-splines of arbitrary degree as well as an associated Hodge operator. The theory is first developed in 1D and then extended to multi-dimension using tensor products. We link our discrete differential forms with the theory of chains and cochains. The spline discrete differential forms are then applied to the numerical solution of Maxwell's equations

Finite Element Hodge for Spline Discrete Differential Forms. Application to the Vlasov-Poisson Equations.

The notion of B-spline based discrete differential forms is recalled and along with a Finite Element Hodge operator, it is used to design new numerical methods for solving the Vlasov-Poisson equations

GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods

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.Comment: 57 Page

Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method

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

Handling the divergence constraints in Maxwell and Vlasov-Maxwell simulations

International audienceThe aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the Vlasov-Maxwell equations and the Maxwell equations with sources. These methods can be classified in two types: field correction methods and sources correction methods. The field correction methods introduce new unknowns in the equations, for which additional boundary conditions are in some cases non trivial to find. The source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete Gauss' law that needs to be defined in accordance with the discretization of the Maxwell propagation operator