Field propagator

ImplicitDiffusion

Advances phi.

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What it solves

Weak, implicit discretization of the diffusion (or heat) equation (can be used as a Poisson solver too).

Find $ϕ \in H^{1}$ such that

\int_{Ω} ψ n_{0} (x) \frac{\partial ϕ}{\partial t} d x + \int_{Ω} \nabla ψ^{⊤} D_{0} (x) \nabla ϕ d x = i \sum \int_{Ω} ψ ρ_{i} (x) d x \forall ψ \in H^{1},

where $n_{0}, ρ_{i} : Ω \to R$ are real-valued functions and $D_{0} : Ω \to R^{3 \times 3}$ is a positive diffusion matrix. Boundary terms from integration by parts are assumed to vanish. The equation is discretized as

(\frac{σ _{1}}{Δ t} M_{n_{0}}^{0} + G^{⊤} M_{D_{0}}^{1} G) ϕ^{n + 1} = \frac{σ _{2}}{Δ t} M_{n_{0}}^{0} ϕ^{n} + \frac{σ _{3}}{Δ t} i \sum (Λ^{0}, ρ_{i})_{L^{2}},

where $M_{n_{0}}^{0}$ and $M_{D_{0}}^{1}$ are WeightedMassOperators and $σ_{1}, σ_{2}, σ_{3} \in R$ are artificial parameters that can be tuned to change the model (see Notes).

Variables it advances

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

Name	Kind	Accepted spaces
`phi`	FEECVariable	`H1`

Options

Set these on the propagator in a parameter file, for example ImplicitDiffusion.Options(sigma_1=...).

Option	Default	Choices	Description
`sigma_1`	`1`	float	Coefficient multiplying the stabilization/mass contribution on the left-hand side. With `divide_by_dt=True` it is interpreted as `sigma_1 / dt` in the assembled linear system.
`sigma_2`	`0`	float	Coefficient multiplying the previous solution contribution `stab_mat * phi^n` on the right-hand side. With `divide_by_dt=True` it is interpreted as `sigma_2 / dt`.
`sigma_3`	`1`	float	Coefficient multiplying the source terms `rho` on the right-hand side. With `divide_by_dt=True` it is interpreted as `sigma_3 / dt`.
`divide_by_dt`	`False`	bool	If `True`, divide `sigma_1`, `sigma_2` and `sigma_3` by the time step passed to `__call__(dt)` before assembling the system.
`stab_mat`	`M0`	`M0M0adId`	Stabilization operator used in the term weighted by `sigma_1` and `sigma_2`.
`diffusion_mat`	`M1`	`M1M1perp`	Diffusion metric in the bilinear form `grad.T @ diffusion_mat @ grad`. You can pass the name of a pre-built operator in `mass_ops` or a custom `WeightedMassOperator` compatible with the codomain of `grad`.
`x0`	`None`	StencilVector	Initial guess for the iterative linear solver.
`solver`	`pcg`	`pcgcg`	Name of the symmetric iterative solver passed to `inverse`.
`precond`	`MassMatrixPreconditioner`	`MassMatrixPreconditionerMassMatrixDiagonalPreconditionerNone`	Name of the preconditioner configuration. Currently this class sets `pc=None` internally, so this option is reserved for compatibility and future extensions.
`solver_params`	`None`	SolverParameters	Iterative-solver controls (for example `tol`, `maxiter`, `verbose`, `info`, `recycle`). If `None`, defaults to `SolverParameters()`.
`filter_params`	`None`	dict	If not None, specifies a filter to the accumulation of a specific variable. Keyed by the `pic_variable` of the corresponding `ParticlesToGrid` source in `rho`.

Used by

No model in the catalogue currently composes this propagator — it is available for building new models.