Field propagator

PoissonAdiabaticGyrokinetic

Advances phi.

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What it solves

Weak discretization of the Poisson equation, with adabatic response of electrons.

Find $ϕ \in H^{1}$ such that

\frac{1}{Z ϵ ^{2}} \int_{Ω} \frac{n _{0}}{T _{0}} ψ ϕ d x + \int_{Ω} \frac{n _{0}}{∣ B _{0} ∣ ^{2}} \nabla ψ^{⊤} D \nabla ϕ d x = i \sum \int_{Ω} ψ ρ_{i} (x) d x \forall ψ \in H^{1},

where $D$ can be an anisotropic (gyrokinetic) diffusion operator (or the identity), $ϵ \in R$ is the gyrokinetic ratio defined in units, and Z the charge number of ions. Boundary terms from integration by parts are assumed to vanish.

The equation is discretized as

(\frac{1}{Z ϵ ^{2}} M_{a}^{0} d + G^{⊤} M_{gyro}^{1} G) ϕ^{n + 1} = i \sum (Λ^{0}, ρ_{i})_{L^{2}},

where $M_{gyro}^{1}$ is the gyrokinetic mass matrix and $S$ is a stabilization matrix.

Variables it advances

The variables this step updates over a time step dt, and the discrete spaces each one accepts.

Name	Kind	Accepted spaces
`phi`	FEECVariable	`H1`

Options

Set these on the propagator in a parameter file, for example PoissonAdiabaticGyrokinetic.Options(stab_mat=...).

Option	Default	Choices	Description
`stab_mat`	`M0ad_withT`	`M0M0adIdM0ad_withT`	Stabilization matrix multiplied by `stab_eps`.
`diffusion_mat`	`M1gyro`	`M1M1perpM1gyro`	Diffusion matrix.
`which_geometry`	`cylindrical`	`cylindricaltoroidal`	Geometry of the problem, determines the meaning of `<\phi>` in the stabilization term.
`x0`	`None`	StencilVector	Initial guess for the iterative linear solver.
`solver`	`pcg`	`pcgcg`	Name of the symmetric iterative solver passed to `inverse`.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Name of the preconditioner configuration. Currently this class inherits the same behavior as `ImplicitDiffusion`, where `pc=None` is used internally.
`solver_params`	`None`	SolverParameters	Iterative-solver controls (for example `tol`, `maxiter`, `verbose`, `info`, `recycle`). If `None`, defaults to `SolverParameters()`.
`filter_params`	`None`	dict	If not None, specifies a filter to the accumulation of a specific variable.

Used by

Each model splits its time step into a sequence of propagators. This one appears in 1 model, at the position shown.

DriftKineticElectrostaticAdiabaticKineticstep 1 of 3 · gc_poisson