Field propagator

VariationalResistivity

Advances s and b.

What it solves

FEEC discretization of the following equations:

find $s \in L^{2}$ and $B \in H (div)$ such that

\int_{Ω} \partial_{t} B \cdot v d x + \int_{Ω} (η + η_{a} (x)) \nabla \times B \cdot \nabla \times v d x = 0 \forall v \in H (div), \int_{Ω} \frac{\partial U}{\partial s} \partial_{t} s q d x - \int_{Ω} (η + η_{a} (x)) ∣\nabla \times B ∣^{2} q d x = 0 \forall q \in L^{2} if using s, \int_{Ω} \frac{1}{γ - 1} \partial_{t} p q d x - \int_{Ω} (η + η_{a} (x)) ∣\nabla \times B ∣^{2} q d x = 0 \forall q \in L^{2} if using p .

With $η_{a} (x) = η_{a} ∣\nabla \times B (x) ∣$

On the logical domain:

\partial_{t} B^- η Δ B^= 0, \int_{Ω^} \partial_{t} (ρ^U (ρ^, s^)) w^\frac{1}{g} d η - \int_{Ω^} (η + η_{a} (η)) ∣ D F^{- T} \nabla ~ \times B^∣^{2} w^d η = 0, if using s, \int_{Ω^} \partial_{t} (\frac{1}{γ - 1} p^) w^\frac{1}{g} d η - \int_{Ω^} (η + η_{a} (η)) ∣ D F^{- T} \nabla ~ \times B^∣^{2} w^d η = 0, if using p .

It is discretized as

\frac{B ^{n + 1} - B ^{n}}{Δ t} + C M_{1}^{- 1} (η M_{1} + η_{a} M_{1} [∣\nabla \times B ∣]) M_{1}^{- 1} C^{T} M_{2} B^{n + 1} = 0, \frac{P ^{3} ( ρ e ( s ^{n + 1} ) - P ^{3} ( ρ e ( s ^{n} ))}{Δ t} - P^{3} ((η + η_{a} (x)) D F^{- T} C ~ \frac{B ^{n + 1} + B ^{n}}{2} \cdot D F^{- T} C ~ B^{n + 1}) = 0, if using s, \frac{1}{γ - 1} \frac{p ^{n + 1} - p ^{n}}{Δ t} - P^{3} ((η + η_{a} (x)) D F^{- T} C ~ \frac{B ^{n + 1} + B ^{n}}{2} \cdot D F^{- T} C ~ B^{n + 1}) = 0, if using p .

where $P^3$ denotes the $L^2$ projection in the last space of the de Rham sequence.

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

Name	Kind	Accepted spaces
`s`	FEECVariable	`L2`
`b`	FEECVariable	`Hdiv`

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

Option	Default	Choices	Description
`model`	`full`	`fullfull_pfull_qlinear_plinear_qdeltaf_q`	Thermodynamic model variant.
`gamma`	`1.66667`	float	Adiabatic index.
`solver`	`pcg`	`pcgcg`	Linear solver for implicit subproblems.
`precond`	`MassMatrixDiagonalPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner used in linear solves.
`solver_params`	`None`	SolverParameters	Linear-solver controls.
`nonlin_solver`	`None`	NonlinearSolverParameters	Nonlinear iteration controls.
`linearize_current`	`False`	bool	If `True`, linearize current terms around background fields.
`eta`	`0`	float	Physical resistivity coefficient.
`eta_a`	`0`	float	Artificial-resistivity coefficient.

Each model splits its time step into a sequence of propagators. This one appears in 7 models, at the position shown.