A fast and accurate numerical implementation of the envelope model for laser–plasma dynamics

D. Terzani and P. Londrillo

Computer Physics Communications 242 (2019) 49–59

Abstract

In laser-driven, plasma wakefield acceleration regimes (LWFA), when relevant scale lengths of the laser envelope and of the driven plasma waves are well separated from the wavelength and frequency of the laser fast oscillating component, a reduced physical model (usually referred to as the envelope model), has been introduced, allowing to formulate the laser–plasma equations in terms of laser cycle-averaged dynamical variables. As a main consequence, physical regimes where this reduced model applies, can be investigated with significant savings of computational resources still assuring comparable accuracy, with respect to standard Particle-In-Cell (PIC) models where all relevant space–time scales have to be resolved. Here we propose a computational framework characterized by two previously unexplored numerical implementations of the envelope model. The first one is based on explicit second order leapfrog integration of the exact wave equation for laser pulse propagation in a laboratory coordinate system in 3D cartesian geometry, replacing the usually quoted representation in an Eulerian frame moving at the speed of light. Since the laser and driven wakefield wave equations in a laboratory frame are advection dominated, we introduce a proper modification of finite differences approximating longitudinal space derivatives, to minimize dispersive numerical errors coming from the discretized advection operators. The proposed implementation, avoiding semi-implicit procedures otherwise required when dealing with a comoving frame, assures significant saving in computational time and ease of implementation for parallel platforms. The associated equation of motion for plasma particles has been integrated, as in standard PIC codes, using the Boris pusher, properly extended to take into account the specific form of the Lorentz force in the envelope model. As a second contribution, a novel numerical implementation of the plasma dynamics equations in the cold-fluid approximation, is presented. The scheme is based on the second-order one-step Adams– Bashforth time integrator coupled to upwind non-oscillatory WENO reconstruction for discretized space derivatives. The proposed integration scheme for the Eulerian fluid equations is equivalent to a leapfrog scheme with an added higher order dissipative truncation errors. It can be used either as a much faster, yet of comparable accuracy, alternative to the PIC representation of plasma particle motion, or even in a hybrid fluid–particle combination when kinetic effects and particle injection and acceleration in a wakefield have to be investigated.