Field propagator

MagnetosonicUniform

Advances n, u and p.

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What it solves

FEEC discretization of the following equations:

find $ρ ~ \in L^{2}, U ~ \in {H (curl), H (div), (H^{1})^{3}}, p ~ \in L^{2}$ such that

\int ρ_{0} \frac{\partial ρ ~}{\partial t} + \nabla \cdot (ρ_{0} U ~) = 0, \frac{\partial U ~}{\partial t} \cdot V d x - \int p ~ \nabla \cdot V d x = 0 \forall V \in {H (curl), H (div), (H^{1})^{3}}, \frac{\partial p ~}{\partial t} + \frac{5}{3} p_{0} \nabla \cdot U ~ = 0 .

time_discret: Crank-Nicolson (implicit mid-point). System size reduction via struphy.linear_algebra.schur_solver.SchurSolver:

[u^{n + 1} - u^{n} p_{i}^{n + 1} - p_{i}^{n}] = \frac{Δ t}{2} [0 - γ K^{3} D (M_{2}^{ρ})^{- 1} D^{⊤} M_{3} 0] [(u^{n + 1} + u^{n}) (p_{i}^{n + 1} + p_{i}^{n})],

where $M_{2}^{ρ}$ is a weighted mass matrix in 2-space, the weight being the MHD equilibirum density $ρ_{0}$ . Furthermore, $K^{3}$ is the basis projection operator given by :

K_{ij k, mn o}^{3} := Π^_{ij k}^{3} [\frac{p ^ _{eq}^{3}}{g} Λ_{mn o}^{3}] .

The solution of the above system is based on the Schur complement.

Decoupled density update:

ρ^{n + 1} = ρ^{n} - \frac{Δ t}{2} Q D (u^{n + 1} + u^{n}) .

Variables it advances

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

Name	Kind	Accepted spaces
`n`	FEECVariable	`L2`
`u`	FEECVariable	`HcurlHdivH1vec`
`p`	FEECVariable	`L2`

Options

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

Option	Default	Choices	Description
`solver`	`pbicgstab`	`pbicgstabbicgstabgmres`	Iterative solver used by `SchurSolver`.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner applied to the `M2n` block.
`solver_params`	`None`	SolverParameters	Iterative-solver controls. If `None`, defaults to `SolverParameters()`.

Used by

Each model splits its time step into a sequence of propagators. This one appears in 1 model, at the position shown.

LinearExtendedMHDuniformFluidstep 3 of 3 · mag_sonic