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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.

Numerical Approximation of Self-Consistent Vlasov Models for Low-Frequency Electromagnetic PhenomenaWe present a new numerical method to solve the Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency electromagnetic phenomena in plasmas, and thus "light waves" are somewhat supressed, which in turn allows the numerical discretization to dispense with the Courant-Friedrichs-Lewy condition on the time step. We construct a numerical scheme based on semi-Lagrangian methods and time splitting techniques. We develop a four-dimensional phase space algorithm for the distribution function while the electromagnetic field is solved on a two-dimensional Cartesian grid. Finally, we present two nontrivial test cases: (a) the wave Landau damping and (b) the electromagnetic beam-plasma instability. For these cases our numerical scheme works very well and is in agreement with analytic kinetic theory.

Hermite Spline Interpolation on Patches for Parallelly Solving the Vlasov-Poisson EquationThis work is devoted to the numerical simulation of the Vlasov equation using a phase space grid. In contrast to Particle-In-Cell (PIC) methods, which are known to be noisy, we propose a semi-Lagrangian-type method to discretize the Vlasov equation in the two-dimensional phase space. As this kind of method requires a huge computational effort, one has to carry out the simulations on parallel machines. For this purpose, we present a method using patches decomposing the phase domain, each patch being devoted to a processor. Some Hermite boundary conditions allow for the reconstruction of a good approximation of the global solution. Several numerical results demonstrate the accuracy and the good scalability of the method with up to 64 processors. This work is a part of the CALVI project.

We study the existence of spatial periodic solutions for nonlinear elliptic equations $- \Delta u \, + \, g(x,u(x)) = 0, \;x \in {\mathbb R}^N$where gis a continuous function, nondecreasing w.r.t. u. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions gare investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations.

We prove the global existence of weak solutions to the Vlasov–Darwin system in R3 for small initial data. The Vlasov–Darwin system is an approximation of the Vlasov–Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov–Maxwell system, the total energy conservation does not provide an L2‐bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov–Darwin system towards the Vlasov–Poisson system. Copyright © 2003 John Wiley & Sons, Ltd.

We prove the global existence of weak solutions to the Vlasov–Darwin system in R3 for small initial data. The Vlasov–Darwin system is an approximation of the Vlasov–Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov–Maxwell system, the total energy conservation does not provide an L2-bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov–Darwin system towards the Vlasov–Poisson system. Copyright © 2003 John Wiley & Sons, Ltd.

A new scheme for solving the Vlasov equation using a phase space grid is proposed. The algorithm is based on the conservation of the flux of particles, and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity. Several numerical results are presented in two- and four-dimensional phase space and the scheme is compared with the semiLagrangian method. This method is almost as accurate as the semiLagrangian one, and the local reconstruction technique is well suited for parallel computation.

The numerical resolution of kinetic equations and, in particular, of Vlasov-type equations is performed most of the time using particle in cell methods which consist in describing the time evolution of the equation through a finite number of particles which follow the characteristic curves of the equation, the interaction with the external and self-consistent fields being resolved using a grid. Another approach consists in computing directly the distribution function on a grid by following the characteristics backward in time for one time step and interpolating the value at the feet of the characteristics using the grid point values of the distribution function at the previous time step. In this report we introduce this last method, which couples the Lagrangian and Eulerian points of view and its use for the Vlasov equation and equations derived from it.

Summary.: In this paper we perform an asymptotic study of the Maxwell equations with respect to the small parameter where is the characteristic velocity associated with the physical problem and is the speed of light. This enables us to derive the quasistatic and Darwin models as respectively first and second order approximations of the Maxwell equations. Moreover, an interpretation of the obtained variational formulations gives us the appropriate boundary conditions for these models.

Abstract: This work deals with the numerical solution of the Vlasov equation, which provides a kinetic description of the evolution of a plasma, and is coupled with Poisson''s equation. A new forward semi-Lagrangian method is developed. The distribution function is updated on a Eulerian grid, and the pseudo-particles located on the mesh 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. [Copyright &y& Elsevier]