Kinetic model

VlasovMaxwellOneSpecies

Vlasov-Maxwell equations for one kinetic species.

Overview

VlasovMaxwellOneSpecies is the fully electromagnetic one-species PIC

model in Struphy. It evolves particles and fields self-consistently and supports an optional control-variate formulation for the field coupling.

This model is appropriate for:

Vlasov equation:

\frac{\partial f}{\partial t} + v \cdot \nabla f + \frac{1}{ε} (E + v \times (B + B_{0})) \cdot \frac{\partial f}{\partial v} = 0

Ampère's law:

- \frac{\partial E}{\partial t} + \nabla \times B = \frac{α ^{2}}{ε} \int_{R^{3}} v f d^{3} v

Faraday's law:

\frac{\partial B}{\partial t} + \nabla \times E = 0

where $Z = - 1$ and $A = 1/1836$ for electrons.

At initial time the weak Poisson equation is solved once to weakly satisfy Gauss' law,

\int_{Ω} \nabla ψ^{⊤} \cdot \nabla ϕ d x E (t = 0) = \frac{α ^{2}}{ε} \int_{Ω} \int_{R^{3}} ψ (f - f_{0}) d^{3} v d x \forall ψ \in H^{1} = - \nabla ϕ (t = 0)

Moreover, it is assumed that

\nabla \times B_{0} = \frac{α ^{2}}{ε} \int_{R^{3}} v f_{0} d^{3} v

where $B_{0}$ is the static equilibrium magnetic field.

The model uses the light speed as reference velocity:

v^= c, E^= B^v^, ϕ^= E^x^.

The species parameters are $α = Ω^_{p} / Ω^_{c}$ and $ε = 1/ (Ω^_{c} t^)$ .

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

Create and initialize a Vlasov-Maxwell model:

from struphy.models import VlasovMaxwellOneSpecies

model = VlasovMaxwellOneSpecies()
model.em_fields.e_field
model.em_fields.b_field

model.kinetic_ions.var