Particle-in-cell modeling of relativistic laser–plasma interaction with the adjustable-damping, direct implicit method

Abstract

a b s t r a c t Implicit particle-in-cell codes offer advantages over their explicit counterparts in that they suffer weaker stability constraints on the need to resolve the higher frequency modes of the system. This feature may prove particularly valuable for modeling the interaction of high-intensity laser pulses with overcritical plasmas, in the case where the electrostatic modes in the denser regions are of negligible influence on the physical processes under study. To this goal, we have developed the new two-dimensional electromagnetic code ELIXIRS (standing for ELectromagnetic Implicit X-dimensional Iterative Relativistic Solver) based on the relativistic extension of the so-called Direct Implicit Method [D. Hewett, A.B. Langdon, Electromagnetic direct implicit plasma simulation, J. Comput. Phys. 72 (1987) 121-155]. Dissipation-free propagation of light waves into vacuum is achieved by an adjustable-damping electromagnetic solver. In the highdensity case where the Debye length is not resolved, satisfactory energy conservation is ensured by the use of high-order weight factors. In this paper, we first derive the electromagnetic direct implicit method as a simplified Newton scheme. Its linear properties are then investigated through numerically solving the relation dispersions obtained for both light and plasma waves, accounting for finite space and time steps. Finally, our code is successfully benchmarked against explicit particle-in-cell simulations for two kinds of physical problems: plasma expansion into vacuum and relativistic laser-plasma interaction. In both cases, we will demonstrate the robustness of the implicit solver for crude discretizations, as well as the gains in efficiency which can be realized over standard explicit simulations.

An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm

“…It is important to note that exact conservation of energy requires the implicit field and particle equations to be updated in a nonlinearly consistent manner every time step. This is distinctly different from previous implicit schemes [16,28,29], which do not feature nonlinear consistency. It is also different from the "energy-conserving" scheme developed by Lewis [25], which is explicit and does not conserves energy exactly with finite ∆t.…”

Asymptotic-Preserving Particle-In-Cell methods for the Vlasov–Maxwell system in the quasi-neutral limit

“…An implicit discretization of the Maxwell equations is needed to avoid a Courant condition similar to (75). The currentJ m+1 h used as source term in the discrete Maxwell-Ampère equation (78) is defined by an approximation of the generalized Ohm law (25). It is crucial to make the electric field implicit in this approximation to ensure the consistency with the quasi-neutral model.…”

“…The existence of KMC waves can be derived from the quasi-neutral equations (with motionless ions) presented in Section 2.3. Neglecting the inertia term ∂ t J and the pressure term ∇ · S in the generalized Ohm law (25) and rewriting it with dimensional variables, we obtain the relation In the two-dimensional setting (116)-(117), if the density does not vary along the x-axis and does vary along the y-axis, the above equation simplifies into a Burgers-like nonlinear hyperbolic equation:…”

Introduction of temporal sub-stepping in the Multi-Level Multi-Domain semi-implicit Particle-In-Cell code Parsek2D-MLMD

“…According to Brackbill and Forslund [41], pushing the stability constraint towards the left part of the inequality may lead to numerical instabilities, among which is the Finite Grid Instability-FGI. Notice however that appropriate countermeasures, such as smoothing [6,9] and higher order shape functions for particles [42], may curb the development of the FGI. Conversely, a grid under-resolved in time may incur accuracy problems, since field variations are not accurately sampled.…”