WENO schemes applied to the quasi-relativistic Vlasov--Maxwell model for laser-plasma interaction

In this paper we focus on WENO-based methods for the simulation of the 1D Quasi-Relativistic Vlasov--Maxwell (QRVM) model used to describe how a laser wave interacts with and heats a plasma by penetrating into it. We propose several non-oscillatory methods based on either Runge--Kutta (explicit) or Time-Splitting (implicit) time discretizations. We then show preliminary numerical experiments

Vlasov laser-plasma interaction simulations with a moving grid

The present work focuses on the numerical resolution of a reduced $1$D Vlasov-Maxwell system introduced recently in the physical literature for studying laser-plasma interaction. This system can be seen as a standard Vlasov equation in which the force term is split into two terms: the classical electrostatic field obtained from the Poisson's equation and a magnetic term evolving through Maxwell's type equations. A semi-Lagrangian code is used to study the interaction of ultrashort electromagnetic pulse with plasma; however during the major part of the simulation, many of the grid points are wasted. We then introduce a dynamic mesh which allows us to consider the part of the phase space where the distribution function is not zero

Numerical Scheme for the One-Dimensional Vlasov-Poisson Equation Using Bi-Orthogonal Spline Wavelets

In this paper, a numerical scheme for solving the Vlasov-Poisson equations is proposed. It is based on bi-orthogonal compactly supported spline wavelets. The interest of these wavelets used in this method is their precision in computations. For solving the Vlasov equation, a Strang splitting in time and a semi-lagrangian method are used. For the Poisson equation, a solver based only on wavelets is presented

A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations

The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability. The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov--Poisson solver show a stabilizing effect of the new bases on the simulations

A "metric" semi-Lagrangian Vlasov-Poisson solver

We propose a new semi-Lagrangian Vlasov-Poisson solver. It employs elements of metric to follow locally the flow and its deformation, allowing one to find quickly and accurately the initial phase-space position $Q(P)$ of any test particle $P$, by expanding at second order the geometry of the motion in the vicinity of the closest element. It is thus possible to reconstruct accurately the phase-space distribution function at any time $t$ and position $P$ by proper interpolation of initial conditions, following Liouville theorem. When distorsion of the elements of metric becomes too large, it is necessary to create new initial conditions along with isotropic elements and repeat the procedure again until next resampling. To speed up the process, interpolation of the phase-space distribution is performed at second order during the transport phase, while third order splines are used at the moments of remapping. We also show how to compute accurately the region of influence of each element of metric with the proper percolation scheme. The algorithm is tested here in the framework of one-dimensional gravitational dynamics but is implemented in such a way that it can be extended easily to four or six-dimensional phase-space. It can also be trivially generalised to plasmas.Comment: 32 pages, 14 figures, accepted for publication in Journal of Plasma Physics, Special issue: The Vlasov equation, from space to laboratory plasma