r"""
Brusselator - Using the `PDE` class
===================================

This example uses the :class:`~pde.pdes.pde.PDE` class to implement the
`Brusselator <https://en.wikipedia.org/wiki/Brusselator>`_ with spatial
coupling,

.. math::

    \partial_t u &= D_0 \nabla^2 u + a - (1 + b) u + v u^2 \\
    \partial_t v &= D_1 \nabla^2 v + b u - v u^2
    
Here, :math:`D_0` and :math:`D_1` are the respective diffusivity and the
parameters :math:`a` and :math:`b` are related to reaction rates.

Note that the same result can also be achieved with a 
:doc:`full implementation of a custom class <pde_brusselator_class>`, which
allows for more flexibility at the cost of code complexity.
"""

from pde import PDE, FieldCollection, PlotTracker, ScalarField, UnitGrid

# define the PDE
a, b = 1, 3
d0, d1 = 1, 0.1
eq = PDE(
    {
        "u": f"{d0} * laplace(u) + {a} - ({b} + 1) * u + u**2 * v",
        "v": f"{d1} * laplace(v) + {b} * u - u**2 * v",
    }
)

# initialize state
grid = UnitGrid([64, 64])
u = ScalarField(grid, a, label="Field $u$")
v = b / a + 0.1 * ScalarField.random_normal(grid, label="Field $v$")
state = FieldCollection([u, v])

# simulate the pde
tracker = PlotTracker(interval=1, plot_args={"vmin": 0, "vmax": 5})
sol = eq.solve(state, t_range=20, dt=1e-3, tracker=tracker)