Coupling propagator

PressureCoupling6D

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 $f$ such that

\int_{Ω} ρ_{0} \frac{\partial U ~}{\partial t} \cdot V d x = - \frac{A _{h}}{A _{b}} \nabla \cdot P ~_{h, ⊥} \cdot V d x \forall V \in {H (curl), H (div), (H^{1})^{3}}, \frac{\partial f _{h}}{\partial t} - (\nabla U ~_{⊥} \cdot v) \cdot \frac{\partial f _{h}}{\partial v} = 0, P ~_{h, ⊥} = \int_{R^{3}} f_{h} v_{⊥} v_{⊥}^{⊤} d^{3} v .

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 - D F^{- ⊤} Λ ˉ^{1} G X ˉ V (M^{n})^{- 1} (M^{n})^{- 1} V^{⊤} (X ˉ)^{⊤} G^{⊤} (Λ ˉ^{1})^{⊤} D F ˉ^{- 1} 0] [M^{n} (u^{n + 1} + u^{n}) W ˉ (V^{n + 1} + V^{n}] .

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	`Particles6D`

Set these on the propagator in a parameter file, for example PressureCoupling6D.Options(ep_scale=...).

Option	Default	Choices	Description
`ep_scale`	`1`	float	Scaling factor applied to energetic-particle pressure coupling terms.
`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 applied to the mass matrix block for `u`.
`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()`.
`use_perp_model`	`True`	bool	If `True`, use the perpendicular pressure-coupling model; otherwise use the full-model kernels.

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