Field propagator

VariationalViscosity

Advances s and u.

What it solves

FEEC discretization of the following equations:

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

\int_{Ω} \partial_{t} (ρ u) \cdot v d x + \int_{Ω} (μ + μ_{a} (x)) \nabla u : \nabla v d x = 0 \forall v \in (H^{1})^{3}, \int_{Ω} \frac{\partial U}{\partial s} \partial_{t} s q d x - μ \int_{Ω} ∣\nabla u ∣^{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 - μ \int_{Ω} ∣\nabla u ∣^{2} q d x = 0 \forall q \in L^{2} if using p .

With $μ_{a} (x) = μ_{a} ∣\nabla u (x) ∣$

On the logical domain:

\int_{Ω^} \partial_{t} (ρ^^{3} u^) \cdot G v^d η + μ \int_{Ω^} \nabla (D F u^) : \nabla (D F v^) \frac{1}{g} d η = 0, \int_{Ω^} \partial_{t} (ρ^e^(ρ^, s^)) w^\frac{1}{g} d η - \int_{Ω^} (μ + μ_{a} (η)) \nabla (D F u^) : \nabla (D F u^) w^d η = 0, if using s, \int_{Ω^} \partial_{t} (\frac{1}{γ - 1} p^) w^\frac{1}{g} d η - \int_{Ω^} (μ + μ_{a} (η)) \nabla (D F u^) : \nabla (D F u^) w^d η = 0, if using p .

It is discretized as

M^{v} [ρ^_{h}^{n}] \frac{u ^{n + 1} - u ^{n}}{Δ t} + ν \sum (G X_{ν}^{v})^{T} (μ M_{0} + μ_{a} M_{0} [∣\nabla u ∣] G X_{ν}^{v} u^{n + 1} = 0, \frac{P ^{3} ( ρ ^ _{h}^{n} U ( ρ ^ _{h}^{n} , s ^ _{h}^{n} )) - P ^{3} ( ρ ^ _{h}^{n} U ( ρ ^ _{h}^{n} , s ^ _{h}^{n + 1} ))}{Δ t} - μ P^{3} (ν \sum D F X_{ν}^{v} \frac{u ^{n + 1} + u ^{n}}{2} \cdot D F X_{ν}^{v} u^{n + 1}) = 0, if using s, \frac{1}{γ - 1} \frac{p ^{n + 1} - p ^{n}}{Δ t} - μ P^{3} (ν \sum D F X_{ν}^{v} \frac{u ^{n + 1} + u ^{n}}{2} \cdot D F X_{ν}^{v} u^{n + 1}) = 0, if using p .

where $P^3$ denotes the $L^2$ projection in the last space of the de Rham sequence and the weights in $M_{0} [∣\nabla u ∣]$ are given by

P^{0} (g ν \sum ∣ (G X_{ν}^{v} u)^{⊤} Λ^{0} ∣^{2}])^{⊤} Λ^{0} .

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 VariationalViscosity.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.
`mu`	`0`	float	Physical viscosity coefficient.
`mu_a`	`0`	float	Artificial-viscosity coefficient.
`alpha`	`0`	float	Optional linear damping/regularization parameter.

Each model splits its time step into a sequence of propagators. This one appears in 8 models, at the position shown.