r"""
Diffusion equation with spatial dependence
==========================================

This example solve the
`Diffusion equation <https://en.wikipedia.org/wiki/Diffusion_equation>`_ with a
heterogeneous diffusivity:

.. math::
    \partial_t c = \nabla\bigr( D(\boldsymbol r) \nabla c \bigr)

using the :class:`~pde.pdes.pde.PDE` class. In particular, we consider
:math:`D(x) = 1.01 + \tanh(x)`, which gives a low diffusivity on the left side of the
domain.

Note that the naive implementation,
:code:`PDE({"c": "divergence((1.01 + tanh(x)) * gradient(c))"})`, has numerical
instabilities. This is because two finite difference approximations are nested. To
arrive at a more stable numerical scheme, it is advisable to expand the divergence, 

.. math::
    \partial_t c = D \nabla^2 c + \nabla D . \nabla c
"""

from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph

# Expanded definition of the PDE
diffusivity = "1.01 + tanh(x)"
term_1 = f"({diffusivity}) * laplace(c)"
term_2 = f"dot(gradient({diffusivity}), gradient(c))"
eq = PDE({"c": f"{term_1} + {term_2}"}, bc={"value": 0})


grid = CartesianGrid([[-5, 5]], 64)  # generate grid
field = ScalarField(grid, 1)  # generate initial condition

storage = MemoryStorage()  # store intermediate information of the simulation
res = eq.solve(field, 100, dt=1e-3, tracker=storage.tracker(1))  # solve the PDE

plot_kymograph(storage)  # visualize the result in a space-time plot