Simulation of an Inductively Coupled Plasma with a Two-Dimensional Darwin Particle-in-Cell Code

Electromagnetic simulation with an explicit algorithm has a severe limitation on the time step due to the large speed of light propagation resulting in the high numerical cost. Fully implicit electromagnetic algorithms do not have this limitation but are more complex to implement. Another option is the Darwin method omitting the electromagnetic wave propagation [1]. The Darwin method separates the electric field into solenoidal (electromagnetic) and irrotational (electrostatic) parts. In this work, we propose a new Darwin scheme for simulation of low-frequency electromagnetic processes in laboratory plasmas. A two-dimensional particle-in-cell code in Cartesian geometry has been developed based on the direct implicit Darwin electromagnetic algorithm described in Ref. 1. The new code has several significant modifications compared to the original algorithm. First, the original method is replaced by a new method based on the equation for the vorticity of the solenoidal electric field. And the linear system of equations in the vorticity method is reliably solved using a standard iterative solver. Second, the electromagnetic fields are defined on staggered grids convenient for electromagnetic simulation. Third, the contribution of collisional scattering is included in calculation of the solenoidal electric fields. Fourth, the code includes several solvers for the self-consistent magnetic field with different boundary conditions. Once one of these methods is selected for a particular simulation, the choice can be verified by checking the energy conservation. A two-dimensional particle-in-cell code has been developed using the modified direct implicit Darwin electromagnetic algorithm described in Ref. 2. The code is a valuable tool for simulation of various electromagnetic effects, for example the inductively coupled plasmas and the electromagnetic plasma waves. The code can be used to design future plasma thrusters. References: [1] M. R. Gibbons and D. W. Hewett, "The Darwin Direct Implicit Particle-in Cell (DADIPIC) Method for Simulation of Low Frequency Plasma Phenomena," J. Comput. Phys. 120, 231–247 (1995). [2] Dmytro Sydorenko, et al., "Improved algorithm for a two-dimensional Darwin particle-in-cell code", Phys. Plasmas 32, 043904 (2025).