Toy model

Maxwell

Maxwell's equations in vacuum for electromagnetic field evolution.

Overview

This model simulates the propagation of electromagnetic waves in vacuum

using Maxwell's equations without sources. It uses a finite element exterior calculus (FEEC) formulation with the electric field in H(curl) and the magnetic field in H(div) spaces.

Use cases

Propagation of electromagnetic waves in vacuum.

Governing equations

PDEs solved by model:

Ampère's law (no current):

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

Faraday's law:

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

Normalization

Velocity and fields are normalized as:

v^= c, E^= c B^

where $c$ is the speed of light.

Discretization

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

struphy.propagators.maxwell_weak_ampere.MaxwellWeakAmpere

Diagnostics

The following scalars are tracked during simulation:

Electric energy: $E_{E} = \frac{1}{2} \int ∣ E ∣^{2} d V$
Magnetic energy: $E_{B} = \frac{1}{2} \int ∣ B ∣^{2} d V$
Total energy: $E_{t o t a l} = E_{E} + E_{B}$

Example

Create and initialize a Maxwell model:

from struphy.models import Maxwell

model = Maxwell()
# Fields are accessible via:
model.em_fields.e_field

model.em_fields.b_field