Coupling propagator

CurrentCoupling5DGradB

Advances u and energetic_ions.

What it solves

FEEC discretization of the following equations:

find $U \in {H (curl), H (div), (H^{1})^{3}}$ and $B \in H (div)$ such that

⎩ ⎨ ⎧ \int ρ_{0} \frac{\partial U ~}{\partial t} \cdot V d x = - \frac{A _{h}}{A _{b}} \iint μ \frac{f ^{vol}}{B _{∥}^{*}} (b_{0} \times \nabla B_{∥}) \times B \cdot V d x d v_{∥} d μ \forall V \in {H (curl), H (div), (H^{1})^{3}}, \frac{\partial f}{\partial t} = \frac{1}{B _{∥}^{*}} [b_{0} \times (U ~ \times B)] \cdot \nabla f .

time_discret: Explicit ('rk4', 'forward_euler', 'heun2', 'rk2', 'heun3').

[u ˙ H ˙] = [0 - G ˉ^{- 1} b ˉ_{0}^{\times} G ˉ^{- 1} B ˉ_{f}^{\times} \frac{1}{B ˉ _{∥}^{* 0}} \frac{1}{g ˉ} (L^{2})^{⊤} (M^{2, n})^{- 1} (M^{2, n})^{- 1} L^{2} \frac{1}{g ˉ} \frac{1}{B ˉ _{∥}^{* 0}} B ˉ_{f}^{\times} G ˉ^{- 1} b ˉ_{0}^{\times} G ˉ^{- 1} 0] [M^{2, n} u \frac{A _{h}}{A _{b}} M ˉ W ˉ \nabla B_{∥}] .

For the detail explanation of the notations, see `2022_DriftKineticCurrentCoupling `_.

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

Name	Kind	Accepted spaces
`u`	FEECVariable	`HcurlHdivH1vec`
`energetic_ions`	PICVariable	—

Set these on the propagator in a parameter file, for example CurrentCoupling5DGradB.Options(ep_scale=...).

Option	Default	Choices	Description
`ep_scale`	`1`	float	Scaling factor applied in particle accumulation.
`algo`	`explicit`	`discrete_gradientexplicit`	Time integration algorithm.
`butcher`	`None`	ButcherTableau	Butcher tableau used in explicit mode. If `None` and `algo="explicit"`, defaults to `ButcherTableau()`.
`u_space`	`Hdiv`	`HcurlHdivH1vec`	FEEC space used for the unknown `u` variable.
`solver`	`pcg`	`pcgcg`	Symmetric iterative solver used for mass-matrix inversions.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner used with the mass matrix.
`solver_params`	`None`	SolverParameters	Iterative-solver controls. If `None`, defaults to `SolverParameters()`.
`filter_params`	`None`	FilterParameters	Particle-to-grid filtering parameters used by the accumulator. If `None`, defaults to `FilterParameters()`.
`dg_solver_params`	`None`	DiscreteGradientSolverParameters	Nonlinear solver controls for discrete-gradient mode. If `None` and `algo="discrete_gradient"`, defaults to `DiscreteGradientSolverParameters()`.

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