Abstract
The $\unicode[STIX]{x1D6FF}f$-PIC method is widely used for electrostatic particle-in-cell (PIC) simulations. Its basic idea is the ansatz $f=f_{0}+\unicode[STIX]{x1D6FF}f$ ($\unicode[STIX]{x1D6FF}f$-ansatz) where the particle distribution function $f$ is split into a usually time-independent background $f_{0}$ and a time-dependent perturbation $\unicode[STIX]{x1D6FF}f$. In principle, it can also be used for electromagnetic gyrokinetic PIC simulations, but the required number of markers can be so large that PIC simulations become impractical. The reason is a decreasing efficiency of the $\unicode[STIX]{x1D6FF}f$-ansatz for the so-called ‘Hamiltonian formulation’ using $p_{\Vert }$ as a dynamic variable. As a result, the density and current moment of the distribution function develop large statistical errors. To overcome this obstacle we propose to solve the potential equations in the symplectic formulation using $v_{\Vert }$ as a dynamic variable. The distribution function itself is still evolved in the Hamiltonian formulation which is better suited for the numerical integration of the parallel dynamics. The contributions from the full Jacobian of phase space, a finite velocity sphere of the simulation domain and a shifted Maxwellian as a background are considered. Special care has been taken at the discretisation level to make damped magnetohydrodynamics (MHD) mode simulations within a realistic gyrokinetic model feasible. This includes devices like e.g. large tokamaks with a small aspect ratio.
Highlights
The δf -PIC method (Dimits & Lee 1993; Parker & Lee 1993) is widely used for electrostatic gyrokinetic particle-in-cell (PIC) simulations to discretise the gyro-centre particle number distribution function fs of a species s
Before we describe the gyrokinetic model in the Hamiltonian formulation we want to introduce the coordinate transformation p (R, v, μ, t) = v + q δA (R, t) m and its inverse v (R, p, μ, t) = p − q δA (R, t), m which links the symplectic with the Hamiltonian formulation
The control variates method in gyrokinetic PIC simulation The δf -PIC method is a control variates method (see Aydemir (1994) and appendix A) with the restriction α = 1 in (A 1), when it comes to the evaluation of the moments of the distribution function
Summary
The δf -PIC method (Dimits & Lee 1993; Parker & Lee 1993) is widely used for electrostatic gyrokinetic particle-in-cell (PIC) simulations to discretise the gyro-centre particle number distribution function fs of a species s It is based on the δf -ansatz fs = f0s + δfs,. The so-called ‘p -formulation’ (Hahm, Lee & Brizard 1988) ( called the Hamiltonian formulation) of the gyrokinetic Vlasov–Maxwell system does not have this problem It suffers from a large statistical variance in the evaluation of the moments of the distribution function. The basic idea of the proposed algorithm is straightforward, the required accuracy of the numerical implementation is very high This is a consequence of the fact that in the MHD limit (k⊥ρ → 0) the ratio between the adiabatic and non-adiabatic parts of the distribution function can be more than three orders of magnitude. (v) Consistent finite element approach for the discretisation of the parallel electric field perturbation in the parallel dynamics
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