from dataclasses import dataclass
from typing import Callable, Literal
from feectools.linalg.stencil import StencilVector
from struphy.io.options import LiteralOptions
from struphy.linear_algebra.multigrid.preconditioner import MultiGridOptions
from struphy.linear_algebra.solver import SolverParameters
from struphy.models.variables import FEECVariable, PICVariable, SPHVariable
from struphy.pic.accumulation.filter import FilterParameters
from struphy.pic.accumulation.particles_to_grid import ParticlesToGrid
from struphy.propagators.implicit_diffusion import ImplicitDiffusion
from struphy.utils.utils import check_option
[docs]
class GyrokineticPoissonSolve(ImplicitDiffusion):
r"""
Weak discretization of the Poisson equation, with adabatic response of electrons.
Find :math:`\phi \in H^1` such that
.. math::
\frac{1}{Z\epsilon^2} \int_\Omega \frac{n_0}{T_0} \psi\, \phi\,\textrm d \mathbf x + \int_\Omega \frac{n_0}{|B_0|²} \nabla \psi^\top \,\mathbb D \nabla \phi \,\textrm d \mathbf x = \sum_i \int_\Omega \psi\, \rho_i(\mathbf x)\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,,
where :math:`\mathbb D` can be an anisotropic (gyrokinetic) diffusion operator (or the identity),
:math:`\epsilon \in \mathbb R` is the gyrokinetic ratio defined in units, and Z the charge number of ions.
Boundary terms from integration by parts are assumed to vanish.
The equation is discretized as
.. math::
\left( \frac{1}{Z\epsilon^2}\,\mathbb M^0_ad + \mathbb G^\top \mathbb M^1_\textnormal{gyro} \mathbb G \right)\, \boldsymbol\phi^{n+1} = \sum_i(\Lambda^0, \rho_i )_{L^2}\,,
where :math:`\mathbb M^1_\textnormal{gyro}` is the gyrokinetic mass matrix
and :math:`\mathbb S` is a stabilization matrix.
Parameters
----------
phi : StencilVector
FE coefficients of the solution as a discrete 0-form.
stab_eps : float
Stabilization parameter multiplied on stab_mat (default=0.0).
stab_mat : str
Name of the stabilizing matrix.
rho : FEECVariable or Callable or ParticlesToGrid or list
(List of) right-hand side source term(s) of the Poisson problem (optional, can be set with
a setter later). See :class:`~struphy.propagators.implicit_diffusion.ImplicitDiffusion` for
the accepted entries; particle sources are passed as
:class:`~struphy.pic.accumulation.particles_to_grid.ParticlesToGrid`.
x0 : StencilVector
Initial guess for the iterative solver (optional, can be set with a setter later).
solver : dict
Parameters for the iterative solver (see ``__init__`` for details).
"""
def __init__(
self,
rho: FEECVariable | Callable | ParticlesToGrid | list = None,
rho_coeffs: float | list = None,
epsilon: float = 1.0,
Z: int = 1,
diagnostic: FEECVariable | None = None,
):
"""
Parameters
----------
see ImplicitDiffusion.__init__ docstring for other parameters.
epsilon : float
Gyrokinetic parameter, appears in gyrokinetic Poisson equation
Z : int
Charge number for ions, appears in gyrokinetic Poisson equation
"""
super().__init__(rho=rho, rho_coeffs=rho_coeffs, diagnostic=diagnostic)
self.epsilon = epsilon
self.Z = Z
[docs]
@dataclass
class Options:
r"""Configuration options for :class:`GyrokineticPoissonSolve`.
Parameters
----------
epsilon : float, default=1.0
Gyrokinetic parameter
Z : float, default=1.0
Charge number of ions
stab_mat : {"M0", "M0ad", "Id"}, default="Id"
Stabilization matrix multiplied by ``stab_eps``.
- ``"M0"``: standard weighted 0-form mass operator.
- ``"M0ad"``: adiabatic-electron weighted 0-form mass operator.
- ``"Id"``: identity operator.
which_geometry: {"cylindrical", "toroidal"}, default="cylindrical"
Geometry of the problem, determines the meaning of `<\phi>` in the stabilization term:
the average over ``eta3`` (cylindrical) or over ``eta2`` and ``eta3`` (toroidal), i.e.
``stab_average`` of :class:`ImplicitDiffusion` is ``"eta3"`` or ``"eta2 eta3"``. The average
is weighted with ``stab_mat`` (for equilibrium density and temperature constant on the
averaged surfaces, the flux-surface average with the volume element), which keeps the
operator symmetric (:class:`~struphy.feec.mass.WeightedAverageProjection`).
diffusion_mat : {"M1", "M1perp", "M1gyro"}, defaults="M1gyro"
Diffusion matrix.
rho : FEECVariable or Callable or ParticlesToGrid or list, default=None
Right-hand side source term(s) of the Poisson problem.
Accepted entries are:
- ``None``: zero source.
- ``FEECVariable`` in ``H1``.
- ``Callable`` to be projected to ``H1`` via ``L2Projector``.
- :class:`~struphy.pic.accumulation.particles_to_grid.ParticlesToGrid`, describing a
particle-to-grid (charge/current) deposition.
- a ``list`` containing any mix of the entries above.
rho_coeffs : float or list, default=None
Multiplicative coefficient(s) applied to ``rho``.
If ``None``, coefficients default to ``1.0`` for all sources.
x0 : StencilVector, default=None
Initial guess for the iterative linear solver.
solver : LiteralOptions.OptsSymmSolver, default="pcg"
Name of the symmetric iterative solver passed to
:func:`psydac.linalg.solvers.inverse`.
precond : LiteralOptions.OptsDiffusionPrecond, default=None
Name of the preconditioner, see :class:`ImplicitDiffusion`. Requires ``solver="pcg"``.
``"StiffnessPreconditioner"`` approximates the operator with ``diffusion_mat``, ``stab_mat``
and the average; ``"MultiGrid"`` is the geometric multigrid V-cycle, with the average
re-built on every coarse level.
precond_params : dict, default=None
Keyword arguments passed to the constructor of the mass-matrix or stiffness
preconditioner, see :class:`ImplicitDiffusion`.
multigrid : MultiGridOptions, default=None
Options of the multigrid preconditioner (if ``precond="MultiGrid"``).
If ``None``, defaults to ``MultiGridOptions()``.
solver_params : SolverParameters, default=None
Iterative-solver controls (for example ``tol``, ``maxiter``,
``verbose``, ``info``, ``recycle``).
If ``None``, defaults to ``SolverParameters()``.
filter_params : dict[PICVariable | SPHVariable, FilterParameters], default=None
If not None, specifies a filter to the accumulation of a specific variable.
Notes
-----
``GyrokineticPoissonSolve`` reuses :class:`ImplicitDiffusion` internals by
enforcing
``sigma_1 = 1 / (Z * epsilon**2)``, ``sigma_2 = 0.0``, ``sigma_3 = 1 / epsilon`` and
``divide_by_dt = False`` in ``allocate``.
"""
# specific literals
OptsStabMat = Literal["M0", "M0ad", "Id", "M0ad_withT"]
OptsDiffusionMat = Literal["M1", "M1perp", "M1gyro"]
OptsGeometry = Literal["cylindrical", "toroidal"]
# propagator options
stab_mat: OptsStabMat = "M0ad_withT"
diffusion_mat: OptsDiffusionMat = "M1gyro"
which_geometry: OptsGeometry = "cylindrical"
x0: StencilVector = None
solver: LiteralOptions.OptsSymmSolver = "pcg"
precond: LiteralOptions.OptsDiffusionPrecond = None
precond_params: dict = None
multigrid: MultiGridOptions = None
solver_params: SolverParameters = None
filter_params: dict[PICVariable | SPHVariable, FilterParameters] = None
# the adiabatic term is physical, not a stabilization: the rhs is not projected
enforce_compatibility: bool = False
def __post_init__(self):
# checks
check_option(self.stab_mat, self.OptsStabMat)
check_option(self.solver, LiteralOptions.OptsSymmSolver)
check_option(self.precond, LiteralOptions.OptsDiffusionPrecond)
check_option(self.which_geometry, self.OptsGeometry)
if self.precond is not None:
assert self.solver == "pcg", f"precond={self.precond!r} requires solver='pcg'."
# defaults
if self.precond_params is None:
self.precond_params = {}
if self.solver_params is None:
self.solver_params = SolverParameters()
if self.multigrid is None:
self.multigrid = MultiGridOptions()
@property
def stab_average(self) -> str:
"""Directions averaged in the stabilization, see ``which_geometry``."""
return {"cylindrical": "eta3", "toroidal": "eta2 eta3"}[self.which_geometry]
def allocate(self):
epsilon = self.epsilon
Z = self.Z
self.options.sigma_1 = 1 / epsilon**2 / self.Z
self.options.sigma_2 = 0.0
self.options.sigma_3 = 1 / epsilon
self.options.divide_by_dt = False
# the stabilization stab_mat @ (I - A) (A from stab_average) is built in ImplicitDiffusion
super().allocate()
@property
def options(self) -> Options:
if not hasattr(self, "_options"):
self._options = self.Options()
return self._options
@options.setter
def options(self, new):
assert isinstance(new, self.Options)
self._options = new
