Darwin Approximation for Numerical Modeling of Rarefied Plasma

A fully kinetic self-consistent model for studying the motion of a rarefied plasma consists of kinetic Vlasov equations for each of the plasma components and Maxwell’s equations for the electromagnetic field. In the non-relativistic limit, the small value of the electron-to-ion mass ratio and the large difference between the speed of light and the characteristic particle velocities lead to the use of a very small time step in the simulation. In this paper, we study the stability issues of a hybrid model 1, in which a kinetic description is used for the positively charged ions only, while the electrons are considered to be a charge-neutralizing fluid, and propose a new, more stable method based on the Darwin approximation 2–4 for simulating slow plasma motion. In the Darwin approximation, the solenoidal part of the displacement current is neglected, making fast electromagnetic wave propagation impossible. Moreover, the hyperbolic system of Maxwell’s equations is transformed into a system of elliptic equations, allowing for the construction of an unconditionally stable numerical scheme. A significant drawback of the Darwin approximation, which is the need to solve eight Poisson equations at each time step, is offset by the possibility to use a significantly larger value of the time step itself. In this paper, we analyze and compare the numerical stability of the hybrid model, the Darwin model, and our new model. This analysis shows that the new method is more stable and, moreover, less expensive than the conventional Darwin scheme.