Field propagator

VariationalMagFieldEvolve

Advances u and b.

What it solves

FEEC discretization of the following equations:

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

\begin{aligned} &\int_\Omega \partial_t (\rho \mathbf u) \cdot \mathbf v\,\textrm d \mathbf x - \int_\Omega \mathbf B \cdot \nabla \times (\mathbf \tilde{B} \times \mathbf v) \,\textrm d \mathbf x = 0 \qquad \forall \, \mathbf v \in (H^1)^3\,, \\[4mm] &\partial_t \mathbf B + \nabla \cdot ( \mathbf \tilde{B} \times \mathbf u ) = 0 \,. \end{aligned}

Where $B ~$ is either $B$ for full-f models, $B_{0}$ for linear models or $B_{0} + B$ for $δ f$ models.

On the logical domain:

\int_{Ω^} \partial_{t} (ρ^^{3} u^) \cdot G v^d η - \int_{Ω^} B^^{2} \cdot G \nabla \times (B^^{2} \times v^) \frac{1}{g} d η = 0, \partial_{t} B^^{2} + \nabla \times (B^^{2} \times u^) = 0 .

It is discretized as

\begin{align} &\mathbb M^v[\hat{\rho}_h^{n}] \frac{ \mathbf u^{n+1}-\mathbf u^n}{\Delta t} - (\mathbb C \hat{\Pi}^{1}[B_h^{n+1}} \cdot \vec{\boldsymbol \Lambda}^v])^\top \mathbb M^2 B^{n+\frac{1}{2}} \big) = 0 ~ , \\[2mm] &\frac{\mathbf b^{n+1}- \mathbf b^n}{\Delta t} + \mathbb C \hat{\Pi}^{1}[\hat{B_h^{n}} \cdot \vec{\boldsymbol \Lambda}^v]] \mathbf u^{n+1/2} = 0 ~ , \end{align}

where weights in the the struphy.feec.basis_projection_ops.BasisProjectionOperator and the struphy.feec.mass.WeightedMassOperator are given by

B^_{h}^{n + 1/2} = (b^{n + 1/2})^{⊤} Λ^{2} \in V_{h}^{2} ρ^_{h}^{n} = (ρ^{n})^{⊤} Λ^{3} \in V_{h}^{3} .

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

Name	Kind	Accepted spaces
`u`	FEECVariable	`H1vec`
`b`	FEECVariable	`Hdiv`

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

Option	Default	Choices	Description
`model`	`full`	`fullfull_plinear`	Magnetic-field evolution model variant.
`solver`	`pcg`	`pcgcg`	Linear solver for implicit substeps.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner used in linear solves.
`solver_params`	`None`	SolverParameters	Linear-solver controls.
`nonlin_solver`	`None`	NonlinearSolverParameters	Nonlinear iteration controls.

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