Field propagator

ShearAlfvenB1

Advances u and b.

What it solves

FEEC discretization of the following equations:

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

\int_{Ω} ρ_{0} \frac{\partial U ~}{\partial t} \cdot V d x = \int_{Ω} B ~ \cdot \nabla \times (B_{0} \times V) d x \forall V \in {H (curl), H (div), (H^{1})^{3}}, \int_{Ω} \frac{\partial B ~}{\partial t} \cdot C d x - \int_{Ω} C \cdot \nabla \times (U ~ \times B_{0}) d x = 0 \forall C \in H (curl) .

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 - M_{1}^{- 1} C^{⊤} T^{2} (M_{2}^{ρ})^{- 1} (M_{2}^{ρ})^{- 1} T^{2}^{⊤} C M_{1}^{- 1} 0] [M_{2}^{ρ} (u^{n + 1} + u^{n}) M_{1} (b^{n + 1} + b^{n})],

where $M_{2}^{ρ}$ is a weighted mass matrix in 2-space, the weight being $ρ_{0}$ , the MHD equilibirum density.

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

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

Set these on the propagator in a parameter file, for example ShearAlfvenB1.Options(solver=...).

Option	Default	Choices	Description
`solver`	`pcg`	`pcgcg`	Solver used for the Schur-reduced system.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner for `M2n` block.
`solver_params`	`None`	SolverParameters	Solver controls for the Schur solve.
`solver_M1`	`pcg`	`pcgcg`	Solver used for inverting `M1`.
`precond_M1`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner for `M1` inversion.
`solver_params_M1`	`None`	SolverParameters	Solver controls for `M1` inversion.

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