Field propagator

VariationalMomentumAdvection

Advances u.

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What it solves

FEEC discretization of the following equations:

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

\int_{Ω} \partial_{t} (ρ u) \cdot v d x - \int_{Ω} ρ u \cdot [u, v] d x = 0 .

On the logical domain:

\int_{Ω^} \partial_{t} (ρ^^{3} u^) \cdot G v^d η - \int_{Ω^} ρ^^{3} u^\cdot G [u^, v^] d η = 0,

which is discretized as

M^{v} [ρ^_{h}^{n}] \frac{u ^{n + 1} - u ^{n}}{Δ t} - (μ \sum (Π^^{0} [u^_{h}^{n + 1/2} \cdot Λ^{1}] G P_{μ} - Π^^{0} [A^_{μ, h}^{1} \cdot Λ^{v}])^{⊤} P_{i}) M^{v} [ρ^_{h}^{n}] u^{n} = 0 .

where $P_{μ}$ stand for the struphy.feec.basis_projection_ops.CoordinateProjector and the weights in the the two struphy.feec.basis_projection_ops.BasisProjectionOperator and the struphy.feec.mass.WeightedMassOperator are given by

u^_{h}^{n + 1/2} = (u^{n + 1/2})^{⊤} Λ^{v} \in (V_{h}^{0})^{3}, A^_{μ, h}^{1} = \nabla P_{μ} ((u^{n + 1/2})^{⊤} Λ^{v})] \in V_{h}^{1}, ρ^_{h}^{n} = (ρ^{n})^{⊤} Λ^{3} \in V_{h}^{3} .

Variables it advances

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

Name	Kind	Accepted spaces
`u`	FEECVariable	`H1vec`

Options

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

Option	Default	Choices	Description
`solver`	`pcg`	`pcgcg`	Linear solver for mass-matrix related solves.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner used in linear solves.
`solver_params`	`None`	SolverParameters	Linear-solver controls.
`nonlin_solver`	`None`	NonlinearSolverParameters	Nonlinear iteration controls (Picard/Newton).

Used by

Each model splits its time step into a sequence of propagators. This one appears in 9 models, at the position shown.