Field propagator

VariationalDensityEvolve

Advances rho and u.

What it solves

FEEC discretization of the following equations:

find $ρ \in L^{2}$ and $u \in (H^{1})^{3}$ such that

\partial_{t} ρ + \nabla \cdot (ρ ~ u) = 0, \int_{Ω} \partial_{t} (ρ u) \cdot v d x + \int_{Ω} (\frac{∣ u ∣ ^{2}}{2} - \frac{\partial ( ρ U ( ρ ))}{\partial ρ}) \nabla \cdot (ρ ~ v) d x = 0 \forall v \in (H^{1})^{3} .

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

In the case of linear model, the second equation is not updated.

On the logical domain:

\partial_{t} ρ^^{3} + \nabla \cdot (ρ^^{3} u^) = 0, \int_{Ω^} \partial_{t} (ρ^^{3} u^) \cdot G v^d η + \int_{Ω^} (\frac{∣ D F u ^ ∣ ^{2}}{2} - \frac{\partial ( ρ ^ ^{3} U )}{\partial ρ ^ ^{3}}) \nabla \cdot (ρ^^{3} v^) d η = 0,

where $U$ depends on the chosen model. It is discretized as

\frac{M ^{v} [ ρ ^ _{h}^{n + 1} ] u ^{n + 1} - M ^{v} [ ρ ^ _{h}^{n} ] u ^{n}}{Δ t} + (D Π^^{2} [ρ_{h}^{n}^Λ^{v}])^{⊤} l^^{3} (\frac{D F u ^ _{h}^{n + 1} \cdot D F u ^ _{h}^{n}}{2} - \frac{ρ ^ _{h}^{n + 1} U ( ρ ^ _{h}^{n + 1} ) - ρ ^ _{h}^{n} U ( ρ ^ _{h}^{n} )}{ρ ^ _{h}^{n + 1} - ρ ^ _{h}^{n}}) = 0, \frac{ρ ^{n + 1} - ρ ^{n}}{Δ t} + D Π^^{2} [ρ_{h}^{n}^Λ^{v}] u^{n + 1/2} = 0,

where $l^^{3} (f)$ denotes the vector representing the linear form $v_{h} \mapsto \int_{Ω^} f (η) v_{h} (η) d η$ , that is the vector with components

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

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

u^_{h}^{k} = (u^{k})^{⊤} Λ^{v} \in (V_{h}^{0})^{3} for k in {n, n + 1/2, n + 1}, ρ^_{h}^{k} = (ρ^{k})^{⊤} Λ^{3} \in V_{h}^{3} for k in {n, n + 1/2, n + 1} .

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

Name	Kind	Accepted spaces
`rho`	FEECVariable	`L2`
`u`	FEECVariable	`H1vec`

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

Option	Default	Choices	Description
`model`	`barotropic`	`pressurelessbarotropicfullfull_pfull_qlineardeltaflinear_qdeltaf_q`	Density/thermodynamic 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.

Each model splits its time step into a sequence of propagators. This one appears in 11 models, at the position shown.