Alain Ghizzo

A Vlasov code for the numerical simulation of stimulated Raman scattering

Numerical simulations of the stimulated Raman scattering are presented using an Eulerian relativistic Vlasov code. Such a code allows a finer resolution in phase space than a particle code and provides a better understanding of the acceleration process for the particles at relativistically high energy. Forward Raman scattering as well as backward Raman scattering are considered to illustrate the possibilities of the Eulerian Vlasov code.

Instability of the time splitting scheme for the one-dimensional and relativistic Vlasov-Maxwell system

The Time Splitting Scheme (TSS) has been examined within the context of the one-dimensional (1D) relativistic Vlasov-Maxwell model. In the strongly relativistic regime of the laser-plasma interaction, the TSS cannot be applied to solve the Vlasov equation. We propose a new semi-Lagrangian scheme based on a full 2D advection and study its advantages over the classical Splitting procedure. Details of the underlying integration of the Vlasov equation appear to be important in achieving accurate plasma simulations. Examples are given which are related to the relativistic modulational instability and the self-induced transparency of an ultra-intense electromagnetic pulse in the relativistic regime.

Two‐stage electron acceleration by simultaneous stimulated Raman backward and forward scattering

The coexistence of stimulated Raman forward and backward scattering of intense electromagnetic radiation, which can occur, for instance, in laser fusion plasmas, is investigated. The simultaneous Raman forward and backward scattering is shown to create an electrostatic field structure which is exceptionally efficient in producing highly relativistic electrons. The mechanism of the electron acceleration is analyzed both by Vlasov–Maxwell simulations with self‐consistent fields and by test particle calculations with prescribed electrostatic fields. The Vlasov–Maxwell simulations reveal that the two plasma waves generated by the backward and forward scattering are spatially separated, and thus form a two‐stage electron ‘‘accelerator.’’

A wavelet-MRA-based adaptive semi-Lagrangian method for the relativistic Vlasov-Maxwell system

In this paper we present a new method for the numerical solution of the relativistic Vlasov-Maxwell system on a phase-space grid using an adaptive semi-Lagrangian method. The adaptivity is performed through a wavelet multiresolution analysis, which gives a powerful and natural refinement criterion based on the local measurement of the approximation error and regularity of the distribution function. Therefore, the multiscale expansion of the distribution function allows to get a sparse representation of the data and thus save memory space and CPU time. We apply this numerical scheme to reduced Vlasov-Maxwell systems arising in laser-plasma physics. Interaction of relativistically strong laser pulses with overdense plasma slabs is investigated. These Vlasov simulations revealed a rich variety of phenomena associated with the fast particle dynamics induced by electromagnetic waves as electron trapping, particle acceleration, and electron plasma wavebreaking. However, the wavelet based adaptive method that we developed here, does not yield significant improvements compared to Vlasov solvers on a uniform mesh due to the substantial overhead that the method introduces. Nonetheless they might be a first step towards more efficient adaptive solvers based on different ideas for the grid refinement or on a more efficient implementation. Here the Vlasov simulations are performed in a two-dimensional phase-space where the development of thin filaments, strongly amplified by relativistic effects requires an important increase of the total number of points of the phase-space grid as they get finer as time goes on. The adaptive method could be more useful in cases where these thin filaments that need to be resolved are a very small fraction of the hyper-volume, which arises in higher dimensions because of the surface-to-volume scaling and the essentially one-dimensional structure of the filaments. Moreover, the main way to improve the efficiency of the adaptive method is to increase the local character in phase-space of the numerical scheme, by considering multiscale reconstruction with more compact support and by replacing the semi-Lagrangian method with more local - in space - numerical scheme as compact finite difference schemes, discontinuous-Galerkin method or finite element residual schemes which are well suited for parallel domain decomposition techniques.

A non-periodic 2D semi-Lagrangian Vlasov code for laser-plasma interaction on parallel computer

For the first time, a 2D electromagnetic and relativistic semi-Lagrangian Vlasov model for a multi-computer environment was developed to study the laser-plasma interaction in an open system. Numerical simulations are presented for situations relevant to the penetration of an ultra-intense laser pulse inside a moderately overdense plasma and the relativistic filamentation instability in the case of an underdense plasma. The Vlasov model revealed a rich variety of phenomena associated with the fast particle dynamics induced by the laser pulse as particle trapping, particle acceleration and relativistic self-induced transparency in overdense plasma. Attention was focused on the efficiency and stability properties on the numerical scheme and implementation facilities on massively parallel computers. Success of the semi-Lagrangian Vlasov model is enhanced by the good conservation of the continuity equation and stability of Maxwell system due to the fine description of the electron distribution function and particularly of the charge density and current density.

Gyrokinetic modeling: A multi-water-bag approach

Predicting turbulent transport in nearly collisionless fusion plasmas requires one to solve kinetic (or, more precisely, gyrokinetic) equations. In spite of considerable progress, several pending issues remain; although more accurate, the kinetic calculation of turbulent transport is much more demanding in computer resources than fluid simulations. An alternative approach is based on a water-bag representation of the distribution function that is not an approximation but rather a special class of initial conditions, allowing one to reduce the full kinetic Vlasov equation into a set of hydrodynamic equations while keeping its kinetic character. The main result for the water-bag model is a lower cost in the parallel velocity direction since no differential operator associated with some approximate numerical scheme has to be carried out on this variable v∥. Indeed, a small bag number is sufficient to correctly describe the ion temperature gradient instability.

Vlasov gyrokinetic simulations of ion‐temperature‐gradient driven instabilities

An Eulerian code that solves the gyrokinetic Vlasov equation in slab geometry is presented. It takes into account the E×B and polarization drifts in the plane perpendicular to the magnetic field, and kinetic effects in the parallel direction. The finite Larmor radius is modelled by a convolution operator. The relation is established between this model and others proposed previously, and they are shown to be equivalent in the limit of long wavelengths and small Larmor radii. The code is applied to investigate ion‐temperature‐gradient modes in the quasi‐neutral regime, with adiabatic electrons. Numerical results are reported for a wide range of parameters, including density and temperature profiles, magnetic field strength, and ion to electron temperature ratio. Normally the plasma evolves towards long wavelength structures, although in some cases (when Landau damping is very weak) more strongly turbulent regimes are observed. Test particles are used to compute diffusion coefficients both in real space and ...

Two-dimensional semi-Lagrangian Vlasov simulations of laser-plasma interaction in the relativistic regime

A semi-Lagrangian two-dimensional fully relativistic Vlasov code for multicomputer environments is developed to study trapped-particle dynamics in phase space induced by relativistic modulational and Raman instabilities. Attention is focused on the efficiency properties of the numerical scheme, which allows a very fine description of particle dynamics in phase space. Vlasov simulations show the appearance of coherent vortex structures as a result of the nonlinear saturation mechanism of the relativistic modulational instability. Growth rates are computed and found to be in good agreement with theoretical values obtained from the dispersion relation by Quesnel et al, [Phys. Plasmas 4, 3358-3368 (1997)] and Guerin et al. [Phys. Plasmas 2, 2807-2814 (1995)]. In the case of coupling between the relativistic modulational instability and two-plasmon decay, stochastic behaviour can be observed due to the competition between different plasmas waves.

The numerical integration of the Vlasov equation possessing an invariant

A new method for the numerical integration of the Vlasov equation is presented, which can be applied whenever its characteristics possess an exact invariant. It consists in expressing the distribution function in terms of the invariant itself. The dimensionality of the phase space is thus reduced of one unity, since the invariant only appears as a label of the Vlasov equation and can be coarsely discretized. This technique is applied to the study of the Kelvin-Helmoltz instability, with a very limited number of invariants. Subsequently an example of ion-temperature-gradient instability is analyzed. Although a larger number of invariants are required to describe the temperature profile, qualitatively correct results can be obtained with fewer invariants. Test particles are used to illustrate stochastic diffusion in the phase space and to calculate the diffusion coefficients.

Parallelization of semi-Lagrangian Vlasov codes

In this report we describe the parallel implementation of semi-Lagrangian Vlasov solvers, which are an alternative to Particle-In-Cell simulations for the numerical investigation of the behaviour of charged particles in their self-consistent electromagnetic fields. The semi-Lagrangian method which couples the Lagrangian and Eulerian point of view is particularly interesting on parallel computers as the solution is computed on grid points the number of which remains constant in time on each processor, unlike the number of particles in PIC simulations, and thus simplifies greatly the parallelisation process.