Field propagator

Hall

Advances b.

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What it solves

FEEC discretization of the following equations:

find $B \in H (curl)$ such that

\int_{Ω} \frac{\partial B ~}{\partial t} \cdot C d x + \frac{1}{ε} \int_{Ω} \nabla \times C \cdot (\frac{\nabla \times B ~}{ρ _{0}} \times B_{0}) d x = 0 \forall C \in H (curl)

time_discret: Crank-Nicolson (implicit mid-point):

b^{n + 1} - b^{n} = \frac{Δ t}{2} M_{1}^{- 1} C^{⊤} M_{2}^{T, ρ} C (b^{n + 1} + b^{n}),

where $M_{2}^{T, ρ}$ is a weighted mass matrix in 2-space, the weight being $\frac{T}{ρ _{0}}$ , the MHD equilibirum density $ρ_{0}$ as a 0-form, and rotation matrix $T v = B_{eq}^{2} \times v,$ . The solution of the above system is based on the Pre-conditioned Biconjugate Gradient Stabilized algortihm (PBiConjugateGradientStab).

Variables it advances

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

Name	Kind	Accepted spaces
`b`	FEECVariable	`Hcurl`

Options

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

Option	Default	Choices	Description
`solver`	`pbicgstab`	`pbicgstabbicgstabgmres`	General iterative solver used for the linear Hall update.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner applied to the `M1` block.
`solver_params`	`None`	SolverParameters	Iterative-solver controls. If `None`, defaults to `SolverParameters()`.

Used by

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

LinearExtendedMHDuniformFluidstep 2 of 3 · hall