High order resolution of the Maxwell–Fokker–Planck–Landau model intended for ICF applications

Abstract

Abstract. A high order, deterministic direct numerical method is proposed for the nonrelativistic 2Dx × 3Dv Vlasov-Maxwell system, coupled with Fokker-Planck-Landau type operators. Such a system is devoted to the modelling of electronic transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It describes the kinetics of plasma physics in the nonlocal thermodynamic equilibrium regime. Strong numerical constraints lead us to develop specific methods and approaches for validation, that might be used in other fields where couplings between equations, multiscale physics, and high dimensionality are involved. Parallelisation (MPI communication standard) and fast algorithms such as the multigrid method are employed, that make this direct approach be computationally affordable for simulations of hundreds of picoseconds, when dealing with configurations that present five dimensions in phase space.

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

“…We have implemented the first order (2.6) and second order (2.8) scheme for the approximation of the Boltzmann equation. Here, the Boltzmann collision operator is discretized by a deterministic method [22][23][24][25]27,52], which gives a spectrally accurate approximation. A classical second order finite volume scheme with slope limiters is applied for the transport operator as sdescribed in Section 4.4.…”

A Vlasov–Fokker–Planck code for high energy density physics

“…Recently, Duclous et al [17] developed a nonrelativistic 2D3V Vlasov-Fokker-Planck-Landau code in cartesian geometry with the electric (magnetic) field in (out of) the plane, which can treat an arbitrary degree of anisotropy and may be used to model instabilities as well as electron transport. This approach is important for the validation of other methods relying on more assumptions, but its applicability is limited by the vast amount of computer memory required to model five dimensions in phase-space.…”

A finite element/Fourier treatment of the Fokker–Planck equation

“…It can be advantageous, however, to think of Eqs. (1), (2), (5), (6), (9), and (10) as a system of coupled partial differential equations requiring simultaneous solutions of the distribution functions and the TR potentials. Such an approach will be the subject of future work.…”

“…A common approach has been to apply finite differencing methods [1][2][3]. More recent work has focused on fast multipole methods [4], spherical harmonic treatments [5], full Fourier spectral treatments [6], wavelet approximations [7], spectral collocation methods [8] and finite volume methods [9]. A finite volume method putforth by Xiong et al [10] resembles the finite element method (FEM) presented here.…”