Fluid model

Poisson

Weak discretization of Poisson's equation with a diffusion matrix, stabilization and an optional time-dependent right-hand side.

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Overview

Poisson is the standalone elliptic field-solve model used for weak

diffusion/Poisson problems. It is also the building block reused by other models for initial electrostatic solves.

Use cases

This model is appropriate for:

elliptic benchmark problems
electrostatic field solves with prescribed source terms
testing weak Poisson discretizations and boundary handling

Governing equations

PDEs solved by model:

Find $ϕ \in H^{1}$ such that

- \nabla \cdot D_{0} (x) \nabla ϕ + n_{0} (x) ϕ = ρ (t, x)

where $n_{0}, ρ (t) : Ω \to R$ are real-valued functions, $ρ (t)$ is parametrized by time $t$ , and $D_{0} : Ω \to R^{3 \times 3}$ is a positive matrix. Boundary terms from integration by parts are assumed to vanish.

Normalization

The coefficient scaling is

D^= n^/ x^^{2}, ρ^= n^.

No dedicated velocity normalization is used.

Discretization

Time integration is performed by the following propagators (in sequence):

struphy.propagators.time_dependent_source.TimeDependentSource (if with_t_dep_source is True)
struphy.propagators.poisson_solve.PoissonSolve

Diagnostics

The following scalars are tracked during simulation:

No default scalar diagnostics are defined by this model.

Example

Create and initialize a Poisson model:

from struphy.models import Poisson

model = Poisson()
model.em_fields.phi

model.em_fields.source