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

Abstract

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.

Introduction

Vlasov models have long been used to study short pulse high intensity laser–plasma interaction where collisions can be ignored. This is the case for parametric instabilities, beat wave, Raman, Brillouin scattering or particle acceleration mechanisms. Since the suggestion by Tajima and Dawson [45] of particle acceleration by means of plasma waves in 1979, various schemes have been proposed to excite large amplitude electron plasma waves (EPW) (theoretically capable of reaching an electric field of the order of GV/m). Such a wave is of interest as a particle accelerator concept since electron plasma wave amplitude can largely exceed the breaking limit of the standard metallic cavity based accelerators, which is of the order of 30 MV/m.

One effective way to produce such a wave uses the forward stimulated Raman scattering (SRS) which is a stimulated decay of incident light wave (pump wave) into a scattered wave and a forward-going plasma wave [24]. The driven version of this process, the plasma beatwave accelerator (PBWA), relies upon the non-linear resonant interaction of two parallel intense laser beams [45] with a frequency difference close to the plasma frequency. In these conditions, the beat of theses two waves resonantly induces a high-phase velocity longitudinal plasma wave which traps and accelerates electrons to relativistic energies. More information can be found for instance in [23].

Intensities above 10¹⁹ W cm⁻² are reached with recently developed pulsed lasers and relativistic plasma wavebreaking can now be experimentally investigated. Particularly, one of the most interesting problems in this domain is the laser propagation through overdense plasma, in which the propagation is classically forbidden (i.e. plasmas having densities above $n_c=1.1\times {10}^{21}{\lambda }_0^2{\text{cm}}^{-3}$, where λ₀ is the laser wavelength in microns). However, at very high intensities, two penetration mechanisms have been considered: relativistic self-induced transparency, and conventional hole boring or forward motion of the critical surface caused by the ponderomotive pressure. Furthermore large amplitude waves can be unstable being subject to parametric instabilities. We will come back to this effect in Sections 6.1 The relativistic parametric instability, 6.2 Self-induced transparency and KEEN waves.

Since collisions can be ignored, the Vlasov–Maxwell system has to be used. Although very interesting analytical results are obtained from more simple fluid models such as non-linear dispersion relations, growth rates, envelope models, Manley–Rowe partition between photons, plasmons or phonons, etc., the importance of resonant wave–particle interaction leads to the use of the full kinetic relativistic Vlasov equation and furthermore to the use of numerical simulation.

However, the simulations based on the well-known Particle-In-Cell (PIC) method, have difficulty in supplying a usefully precise description of the electron acceleration process. This is because the PIC codes lack enough particles to display the detailed phase-space structure of the distribution function which is often obtained in those regions of phase-space where particle and phase velocities are comparable and where trapping occurs. For some work related to laser–plasma interaction and other tasks, using PIC code, we refer to [30], [46], [10].

On the other hand, direct solution of the Vlasov partial differential equation itself on a phase-space grid (the so called Vlasov codes) have been found to be a powerful tool for studying in details the particle dynamics due to the very fine resolution in phase-space [11], [43], [41], [8], [9]. For previous work concerning laser–plasma interaction using Vlasov codes we refer to [37], [38], [4]. Vlasov simulations are slowly introduced in place of Lagrangian PIC models for two main reasons: the lack of numerical noise and the fine resolution in velocity space, provided that the dimension of velocity space is as low as possible. This efficiency (in terms of accuracy and computational time) is however lost when filamentation takes place in phase-space which then requires an important increase of the total number of points of the phase-space grid to follow this filamentation. Usual semi-Lagrangian Vlasov models are not well adapted to describe the Dirac-like distribution functions or fine phase-space filaments which characterize, for instance, the particle acceleration in laser–plasma interaction or the well-known phase-space filamentation in velocity of the Vlasov equation. Much progress is then expected from new adaptive time dependent grids with small mesh size in the non-zero regions and large mesh size for the sparse regions. This approach bears the promise to improve Vlasov solvers in terms of performance and accuracy. In the present work we introduce a two-dimensional phase-space mesh which can be refined or derefined adaptively in time. For this purpose we use a technique based on multiresolution analysis which is in the same spirit as the methods developed in [5], [13], [34]. The method was first conceived in [6] for the Vlasov–Poisson system, then used in the context of beam physics in [31], [7], and parallelized with optimized data structure in [35]. In [32] the same method allows to conserve moments up to any order by using the lifting method introduced in [44].

Let us make clear however, that 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.