Implicit–explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning

A high-order implicit–explicit additive Rung–Kutta time integrator is implemented in a particle-in-cell method based on a high-order discontinuous Galerkin Maxwell solver for the simulation of plasmas. The method enforces Gauss law by a hyperbolic divergence cleaner that transports divergence out of the computational domain at several times the speed of light. The stiffness in the field equations induced by this high transport speeds is alleviated by an implicit time integration, while an explicit time integration ensures a computationally efficient particle update. Simulations on a plasma wave and a Weibel instability show that the implicit–explicit solver is computationally efficient, allowing for computations with high divergence transport speeds that ensure an accurate representation of the governing plasma equations. The high-order method only requires two time steps per plasma wave period. Numerical instability appears when the time step exceeds the plasma frequency time scale. A divergence transport speed of approximately ten times the speed of light is shown to be optimal, since it combines an accurate representation of Gauss law with a small influence of numerical noise on the solution.