Coupling propagator

CurrentCoupling5DCurlb

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 \frac{f ^{vol}}{B _{∥}^{*}} v_{∥}^{2} (\nabla \times b_{0}) \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 _{∥}^{*}} v_{∥} (\nabla \times b_{0}) \cdot (B \times U ~) \nabla_{v_{∥}} f .

time_discret: Crank-Nicolson (implicit mid-point). System size reduction via struphy.linear_algebra.schur_solver.SchurSolver:

[u^{n + 1} - u^{n} V_{∥}^{n + 1} - V_{∥}^{n}] = \frac{Δ t}{2} 0 \frac{1}{B ˉ _{∥}^{* 0}} {b ˉ_{0}^{\nabla \times} (B ˉ_{f}^{\times})^{⊤} V ˉ_{∥} \frac{1}{g ˉ}} \cdot_{vector} {\frac{1}{g ˉ} (L^{2})^{⊤}} (M^{2, n})^{- 1} - (M^{2, n})^{- 1} {L^{2} \frac{1}{g ˉ}} \cdot_{vector} {b ˉ_{0}^{\nabla \times} (B ˉ_{f}^{\times})^{⊤} V ˉ_{∥} \frac{1}{g ˉ}} \frac{1}{B ˉ _{∥}^{* 0}}) 0 [(M^{2, n})^{- 1} (u^{n + 1} + u^{n}) \frac{A _{h}}{A _{b}} W (V_{∥}^{n + 1} + V_{∥}^{n})] .

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 CurrentCoupling5DCurlb.Options(ep_scale=...).

Option	Default	Choices	Description
`ep_scale`	`1`	float	Scaling factor applied to energetic-particle contributions in the accumulation kernel.
`u_space`	`Hdiv`	`HcurlHdivH1vec`	FEEC space used for the unknown `u` variable.
`solver`	`pcg`	`pcgcg`	Symmetric iterative solver used by `SchurSolver`.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner used for the FEEC mass matrix block.
`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()`.

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