| The progress made in the last ten years (from experimental, technological and theoretical points of view) in ultra-high laser intensity laser technology and laser wakefield acceleration of electrons allows one to consider in the near future electron acceleration up to several tens of GeV energy using characteristic plasma lengths of tens of centimeters to meters. Laser-plasma interactions have been successfully described since the beginning of the 80's using Particle-In-Cell (PIC) codes, which obtain an approximate characteristic solution of the Vlasov--Maxwell equation. The convective part of the Vlasov-equation being a source of numerical difficulties, the central technique in the PIC method consists in replacing the solution of the partial differential Vlasov equation by the solution of ordinary differential equations for the motion of macro-particles (the macro-particles trajectories in phase-space being the characteristics of the Vlasov-equation). PIC codes are simple, robust and scalable. However, the introduction of macro-particles is responsible for strong numerical shot-noise which can lead to unphysical and unexpected effects if not maintained at a low reasonable low level. Moreover, PIC codes very often use an explicit scheme to solve for the electromagnetic fields, leading to the well-known CFL stability condition. Being limited by the size of the time and spatial steps, realistic and physically reasonable simulations of electron acceleration over a few centimeters to a few meters of plasma becomes unaffordable in terms of computation time. The constraint on the time step can be relaxed using implicit PIC schemes, but the problem of numerical shot-noise remains. Because of the numerical difficulties to solve the convective term of the Vlasov-equation and the robustness and ease of implementation of PIC codes, the phase-space grid methods have received little attention for describing laser-plasma interaction and has been limited to low-dimensional problems. However, this method is theoretically noiseless and can be of great interest in application where noise is of crucial importance and a fine grained phase-space resolution is needed (for example, when studying particles trapping). In the scope of electron acceleration in under-dense plasmas over a long distance (cm to m), we present in this poster the development of an implicit 1D1P Vlasov Solver using a phase-space grid method. We will present the numerical scheme and its implementation and several related issues (stability of the method and conservation laws).This solver is the first step toward a multi-dimensional Implicit Vlasov Solver for the description of the laser-plasma interaction. |
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