Finite-particle instability in the gyrokinetic delta-f PIC algorithm*

The mathematical form of the gyrokinetic delta-f particle-in-cell algorithm (GK-δf-PIC), as distinct from traditional PIC analysis (Ref. 1), was performed in Ref. 2. There, in addition to a finite-particle instability, an unconditional numerical instability was highlighted and its properties discussed. Additional analyses in Refs. [3] and [4] clarified that the instability, as converged in time step and particle number, is an aliasing (finite-mesh) instability with peculiar properties due to spatial anisotropy in gyrokinetics. These latter two analyses were in the infinite particle limit, where it was assumed that a sufficient number of particles are used to approximate the continuous distribution function arbitrarily closely, albeit with fields defined on a discrete mesh.

Here a complementary analysis is presented to illuminate the remaining instability studied in Ref. 2: that due to insufficient particles. In this analysis, the mesh is eliminated, but finite-particle effects are maintained. It is found that Fourier modes are linearly coupled to each other via particle weights. The diagonal terms are physical and remain in the limit of infinite particle number, and in this case the physical dispersion relation is recovered. With finite number of particles, off-diagonal coupling terms are nonzero. With sufficiently low number of particles (which can still be quite large for some systems), it is shown that this coupled system is unstable. Properties of this finite-particle instability, implications for more complicated systems, and mitigation methods will be discussed.

[1] Birdsall and Langdon. Plasma Physics via Computer Simulation (2004)

[2] Wilkie and Dorland. PoP 23:052111 (2016)

[3] Sturdevant and Parker. J. Comp. Phys. 305:647 (2016)

[4] McMillan. PoP 27:052106 (2020)