Field propagator

VariationalPBEvolve

Advances p, u and b.

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What it solves

FEEC discretization of the following equations:

find $u \in (H^{1})^{3}$ , $p \in L^{2}$ and $B \in H (div)$ such that

\int_{Ω} \partial_{t} (ρ u) \cdot v d x - \int_{Ω} B \cdot \nabla \times (B ~ \times v) - \int_{Ω} \frac{1}{γ - 1} (\nabla \cdot (p ~ v)) d x = 0 \forall v \in (H^{1})^{3}, \partial_{t} B + \nabla \cdot (B ~ \times u) = 0, \partial_{t} p + \nabla \cdot (p ~ u) + (γ - 1) p ~ \nabla \cdot u = 0 .

Where $B ~$ (resp. $p ~$ ) is either $B$ (resp. $p$ ) for full-f models, $B_{0}$ (resp. $p_{0}$ ) for linear models or $B_{0} + B$ (resp. $p_{0} + p$ ) 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} - \frac{g}{γ - 1} \nabla \cdot (p^v^) d η = 0, \partial_{t} B^^{2} + \nabla \times (B^^{2} \times u^) = 0, \partial_{t} p^+ \nabla \cdot (p^u^) + (γ - 1) p^\nabla \cdot G u^= 0

It is discretized as

M^{v} [ρ^_{h}^{n}] \frac{u ^{n + 1} - u ^{n}}{Δ t} - (C Π^^{1} [B_{h}^{n + \frac{1}{2}}^\cdot Λ^{v}])^{⊤} M^{2} B^{n + \frac{1}{2}} = 0, - (D Π^^{2} [p_{h}^{n + \frac{1}{2}}^\cdot Λ^{v}])^{⊤} l^^{3} (\frac{g}{γ - 1}) \frac{b ^{n + 1} - b ^{n}}{Δ t} + C Π^^{1} [B_{h}^{n + \frac{1}{2}}^\cdot Λ^{v}]] u^{n + 1/2} = 0, \frac{p ^{n + 1} - p ^{n}}{Δ t} + (D Π^^{2} [p_{h}^{n + \frac{1}{2}}^\cdot Λ^{v}]] + (γ - 1) Π^^{3} [p_{h}^{n + \frac{1}{2}}^\cdot Λ^{3}] D U^{v}) u^{n + 1/2} = 0,

with

l^^{3} (f)_{ij k} = \int_{Ω^} f Λ_{ij k}^{3} d η

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

and $U^{v}$ is struphy.feec.basis_projection_ops.BasisProjectionOperators.

Variables it advances

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

Name	Kind	Accepted spaces
`p`	FEECVariable	`L2`
`u`	FEECVariable	`H1vec`
`b`	FEECVariable	`Hdiv`

Options

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

Option	Default	Choices	Description
`model`	`full_p`	`full_plineardeltaf`	Pressure-magnetic coupled model variant.
`gamma`	`1.66667`	float	Adiabatic index.
`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.

Used by

Each model splits its time step into a sequence of propagators. This one appears in 3 models, at the position shown.