Field propagator

VariationalEntropyEvolve

Advances s and u.

What it solves

FEEC discretization of the following equations:

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

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

On the logical domain:

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

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

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

\hat{\mathbf{u}}_h^{k} = (\mathbf{u}^{k})^\top \vec{\boldsymbol \Lambda}^v \in (V_h^0)^3 \, \text{for k in} \{n, n+1/2, n+1\}, \qquad \hat{s}_h^{k} = (s^{k})^\top \vec{\boldsymbol \Lambda}^3 \in V_h^3 \, \text{for k in} \{n, n+1/2, n+1\} \qquad \hat{\rho}_h^{n} = (\rho^{n})^\top \vec{\boldsymbol \Lambda}^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
`s`	FEECVariable	`L2`
`u`	FEECVariable	`H1vec`

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

Option	Default	Choices	Description
`model`	`full`	`full`	Entropy-evolution model selection.
`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 3 models, at the position shown.