Coupling propagator

EfieldWeightsCoupling

Advances e and ions.

What it solves

Solves the following substep

\frac{\partial E}{\partial t} = - \frac{α ^{2}}{ε} \int v f_{1} d v, \frac{\partial f _{1}}{\partial t} = \frac{1}{v _{th}^{2} ε} E \cdot v f_{0},

which after discretization and in curvilinear coordinates reads

\frac{d}{d t} w_{p} \frac{d}{d t} M_{1} e = \frac{f _{0, p}}{s _{0, p}} \frac{1}{v _{th}^{2} ε} [D F^{- T} (Λ^{1})^{T} e] \cdot v_{p}, = - \frac{α ^{2}}{ε} \frac{1}{N} p \sum w_{p} Λ^{1} \cdot (D F^{- 1} v_{p}) .

This is solved using the Crank-Nicolson method

[M_{1} (e^{n + 1} - e^{n}) W^{n + 1} - W^{n}] = \frac{Δ t}{2} [0 W - E 0] [e^{n + 1} + e^{n} V^{n + 1} + V^{n}],

where

E W = \frac{α ^{2}}{ε} \frac{1}{N} Λ^{1} \cdot (D F^{- 1} v_{p}), = \frac{f _{0, p}}{s _{0, p}} \frac{1}{v _{th}^{2} ε} (D F^{- 1} v_{p}) \cdot (Λ^{1})^{T},

based on the struphy.linear_algebra.schur_solver.SchurSolver.

The accumulation matrix and vector assembled in struphy.pic.accumulation.particles_to_grid.Accumulator are

B C = EW, B y_{n} = E W .

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

Name	Kind	Accepted spaces
`e`	FEECVariable	`Hcurl`
`ions`	PICVariable	`Particles6DDeltaFParticles6D`

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

Option	Default	Choices	Description
`alpha`	`1`	float	Coupling scaling factor used in the electric-field equation.
`kappa`	`1`	float	Coupling scaling factor used in the particle-weight update.
`solver`	`pcg`	`pcgcg`	Symmetric iterative solver used by `SchurSolver`.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Preconditioner applied to the electric-field mass matrix block.
`solver_params`	`None`	SolverParameters	Iterative-solver controls. If `None`, defaults to `SolverParameters()`.

Each model splits its time step into a sequence of propagators. This one appears in 2 models, at the position shown.