Electromagnetic gyrokinetic PIC simulation with an adjustable control variates method

In the last decade, it became clear that electromagnetic (gyro)kinetic particle-in-cell (PIC) simulations are very demanding in respect to numerical methods and the number of markers used. The Monte Carlo discretization of the gyrokinetic equations leads to a severe signal-to-noise problem: the statistical representation of the physically irrelevant but numerically dominant adiabatic current causing a high statistical noise level. The corresponding inaccuracy problem is very pronounced at high plasma β and/or small perpendicular wave numbers k⊥.We derive several numerical schemes to overcome the problem using an adjustable control variates method which adapts to the dominant adiabatic part of the gyro-center distribution function. We have found that the inaccuracy problem is also present in the quasi-neutrality equation as a consequence of the p∥-formulation [T.S. Hahm, W.W. Lee, A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Phys. Fluids 31 (1988) 1940]. For slab simulations in the magnetohydrodynamic (MHD) limit k⊥→0k⊥→0, the number of markers can be reduced by more than four orders of magnitude compared to a conventional δf scheme.The derived schemes represent first steps on a road to fully adaptive control variates method which can significantly reduce the inherent statistical noise of PIC codes.