(Left) Solid line indicates the scaling of the leading order error, averaged over all grid points, of the conservative finite difference scheme for a Gauss–Legendre grid (the grid used in GS2). The slope of the dotted line corresponds to a first order scheme. (Right) Factor by which the conservative finite difference scheme of Eq. (43) amplifies the true collision operator amplitude at the boundaries of the Gauss–Legendre grid.
Plots showing evolution of the perturbed local density, parallel momentum, and energy over fifty collision times. Without the conserving terms (9)–(11), both parallel momentum and energy decay significantly over a few collision times (long dashed lines). Inclusion of conserving terms with the conservative scheme detailed in Sec. III leads to exact moment conservation (solid lines). Use of a nonconservative scheme leads to inexact conservation that depends on grid spacing (short dashed lines).
Plot of the evolution of entropy generation for the homogeneous plasma slab over 20 collision times. Our initial distribution in velocity space is random noise, and we use a grid with 16 pitch angles and 8 energies. The entropy generation rate is always non-negative and approaches zero in the long-time limit.
Evolution of for the electromagnetic plasma slab with , , and . Inclusion of the ion drag term in the electron-ion collision operator leads to the theoretically predicted damping rate for the parallel current given in Eq. (50). Without the ion drag term, the parallel current decays past zero (at ) and converges to a negative value as the electron flow damps to zero.
Evolution of perturbed parallel flow for the electromagnetic plasma slab with , , and . Without inclusion of the ion drag term in Eq. (12), the electron flow is erroneously damped to zero (instead of to the ion flow).
Damping rate of the slow mode for a range of collisionalities spanning the collisionless to strongly collisional regimes. Dashed lines correspond to the theoretical prediction for the damping rate in the collisional and collisionless limits. The solid line is the result obtained numerically with GS2. Vertical dotted lines denote approximate regions (collisional and collisionless) for which the analytic theory is valid.
Evolution of ion particle and heat fluxes for an electrostatic, two-species Z-pinch simulation. We are considering and . The particle flux is indicated by the solid line and is given in units of . The heat flux is indicated by the dashed line and is given in units of . We see that a steady state is achieved for both fluxes without artificial dissipation.
Linear growth rate spectrum of the entropy mode in a Z pinch for , where is major radius and is density gradient scale length. The solid line is the collisionless result, and the other lines represent the result of including collisions. The short dashed line corresponds to using only the Lorentz operator; the dotted line corresponds to the full model collision operator without conserving terms; and the long dashed line corresponds to the full model collision operator with conserving terms. All collisional cases were carried out with .
Sherman–Morrison variable definitions for Lorentz and energy diffusion operator equations
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