A particle method for the homogeneous Landau equation

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains all the important properties of the Landau operator, namely the conservation of mass, momentum and energy, and the decay of entropy. We illustrate our new method by showing its performance in several test cases including the physically relevant case of the Coulomb interaction. The comparison to the exact solution and the spectral method is strikingly good maintaining 2nd order accuracy. Moreover, an efficient implementation of the method via the treecode is explored. This gives a proof of concept for the practical use of our method when coupled with the classical PIC method for the Vlasov equation.Comment: 27 pages, 14 figures, debloated some figures, improved explanations in sections 2, 3, and

A conservative Galerkin solver for the quasilinear diffusion model in magnetized plasmas

The quasilinear theory describes the resonant interaction between particles and waves with two coupled equations: one for the evolution of the particle probability density function (pdf), the other for the wave spectral energy density (sed). In this paper, we propose a conservative Galerkin scheme for the quasilinear model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry. We construct an unconditionally conservative weak form, and propose a discretization that preserves the unconditional conservation property, by “unconditional” we mean that conservation is independent of the singular transition probability. The discrete operators, combined with a consistent quadrature rule, will preserve all the conservation laws rigorously. The technique we propose is quite general: it works for both relativistic and non-relativistic systems, for both magnetized and unmagnetized plasmas, and even for problems with time-dependent dispersion relations. We represent the particle pdf by continuous basis functions, and use discontinuous basis functions for the wave sed, thus enabling the application of a positivity-preserving technique. The marching simplex algorithm, which was initially designed for computer graphics, is adopted for numerical integration on the resonance manifold. We introduce a semi-implicit time discretization, and discuss the stability condition. In addition, we present numerical examples with a “bump on tail” initial configuration, showing that the particle-wave interaction results in a strong anisotropic diffusion effect on the particle pdf.</p