Semi-implicit scheme for multi-species collision operators in Tokamak Plasma Simulations

Abstract

A second-order semi-implicit time integration method has been developed for solving the linearized collision operator in multi-species plasmas. A key feature of this method is that it treats the collision operator as a single entity, avoiding the operator splitting between its test particle and field particle components. This scheme employs an implicit trapezoidal time integration scheme for the isothermal test particle part (including pitch-angle scattering and energy diffusion) with a finite volume discretization in (v_∥, μ) velocity coordinates, while the non-isothermal model term and field particle part are treated explicitly. This approach avoids the species cross-coupling required by fully implicit schemes, enabling the collision term to be computed efficiently with a banded/sparse-matrix solver. The method has been implemented in the gyrokinetic semi-Lagrangian code NLT and verified through multi-species relaxation tests and neoclassical transport simulations. Numerical benchmarks against explicit methods confirm its robustness, achieving order-of-magnitude improvements in the allowable time-step size, particularly in simulations of electron-ion collisions. Furthermore, the numerical discretization rigorously preserves particle number, momentum, and energy conservation, maintains the self-adjoint property of the collision operator, and satisfies Boltzmann’s H-theorem.