An improved control-variate scheme for particle-in-cell simulations with collisions

Particle-in-cell methods combined with a δf approach constitute an established and powerful method for simulating collisionless kinetic equations in e.g. plasma physics. Including collisions in such simulations requires a modified approach leading to a two-weight scheme, which has the drawback of giving a statistical error that increases with time. As in the collisionless case, this scheme can be interpreted as an application of an ordinary control variate. Using an enhanced control variate approach, an improved scheme is constructed. This approach has been applied to a model problem with the result of a much better behaviour of the error, which, instead of growing indefinitely, becomes bounded by the error of a full-f scheme. In addition, the application of the enhanced control variate is illustrated for a collisionless simulation of ITG turbulence. Here it can be used both as a diagnostic tool and as a means to eliminate the spurious violation of particle number conservation inherent to δf simulations due to statistical noise.