Field propagator

ShearAlfvenCurrentCoupling5D

Advances u and b.

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

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

[u^{n + 1} - u^{n} b^{n + 1} - b^{n}] = \frac{Δ t}{2} . [0 - C T^{2} (M^{2, n})^{- 1} (M^{2, n})^{- 1} T^{2}^{⊤} C^{⊤} 0] M^{2, n} (u^{n + 1} + u^{n}) M_{2} (b^{n + 1} + b^{n}) + \sum_{k}^{N_{p}} ω_{k} μ_{k} b^_{0}^{1} (η_{k}) \cdot (\frac{1}{g ( η _{k} )} Λ^{2} (η_{k})),

where $\mathcal{T}^2 = \hat \Pi \left[\frac{\tilde{\mathbf B}^2}{\sqrt{g} \times \vec \Lambda^2\right]$ and $M^{2, n}$ is a struphy.feec.mass.WeightedMassOperators being weighted with $ρ_{eq}$ , the MHD equilibirum density.

Variables it advances

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

Name	Kind	Accepted spaces
`u`	FEECVariable	`HcurlHdivH1vec`
`b`	FEECVariable	`Hdiv`

Options

Set these on the propagator in a parameter file, for example ShearAlfvenCurrentCoupling5D.Options(ep_scale=...).

Option	Default	Choices	Description
`ep_scale`	`1`	float	Scaling factor for particle accumulation terms.
`u_space`	`Hdiv`	`HcurlHdivH1vec`	FEEC space used for velocity variable `u`.
`algo`	`implicit`	`implicitexplicit`	Time stepping algorithm.
`solver`	`pcg`	`pcgcg`	Symmetric iterative solver.
`precond`	`MassMatrixDiagonalPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner for mass-matrix block.
`solver_params`	`None`	SolverParameters	Solver controls.
`filter_params`	`None`	FilterParameters	Particle-to-grid filtering parameters.
`butcher`	`None`	ButcherTableau	Explicit Runge-Kutta tableau for `algo="explicit"`.
`nonlinear`	`True`	bool	If `True`, include nonlinear TB operator updates.

Used by

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

LinearMHDDriftkineticCCHybridstep 3 of 7 · shearalfen_cc5d