\partial_t \phi = 6 \phi \partial_x \phi - \partial_x^2 \phi
which we implement using a custom PDE class below.
"""
from math import pi
from pde import CartesianGrid, MemoryStorage, PDEBase, ScalarField, plot_kymograph
class KortewegDeVriesPDE(PDEBase):
""" Korteweg-de Vries equation """
def evolution_rate(self, state, t=0):
""" implement the python version of the evolution equation """
assert state.grid.dim == 1 # ensure the state is one-dimensional
grad = state.gradient("natural")[0]
return 6 * state * grad - grad.laplace("natural")
# initialize the equation and the space
grid = CartesianGrid([[0, 2 * pi]], [32], periodic=True)
state = ScalarField.from_expression(grid, "sin(x)")
# solve the equation and store the trajectory
storage = MemoryStorage()
eq = KortewegDeVriesPDE()
eq.solve(state, t_range=3, tracker=storage.tracker(0.1))
# plot the trajectory as a space-time plot
plot_kymograph(storage)
from math import pi
from pde import (CartesianGrid, ScalarField, PDEBase, MemoryStorage,
plot_kymograph)
class KortewegDeVriesPDE(PDEBase):
""" Korteweg-de Vries equation
See https://en.wikipedia.org/wiki/Korteweg–de_Vries_equation
"""
def evolution_rate(self, state, t=0):
""" implement the python version of the evolution equation """
assert state.grid.dim == 1 # ensure the state is one-dimensional
grad = state.gradient('natural')[0]
return 6 * state * grad - grad.laplace('natural')
# initialize the equation and the space
grid = CartesianGrid([[0, 2 * pi]], [32], periodic=True)
state = ScalarField.from_expression(grid, "sin(x)")
eq = KortewegDeVriesPDE()
# solve the equation and store the trajectory
storage = MemoryStorage()
eq.solve(state, t_range=3, tracker=['progress', storage.tracker(.1)])
# plot the trajectory as a space-time plot
plot_kymograph(storage, show=True)
This example implements a complex PDE using the :class:`~pde.pdes.pde.PDE`. We here
chose the `Schrödinger equation <https://en.wikipedia.org/wiki/Schrödinger_equation>`_
without a spatial potential in non-dimensional form:
.. math::
i \partial_t \psi = -\nabla^2 \psi
Note that the example imposes Neumann conditions at the wall, so the wave packet is
expected to reflect off the wall.
"""
from math import sqrt
from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph
grid = CartesianGrid([[0, 20]], 128, periodic=False) # generate grid
# create a (normalized) wave packet with a certain form as an initial condition
initial_state = ScalarField.from_expression(
grid, "exp(I * 5 * x) * exp(-(x - 10)**2)")
initial_state /= sqrt(initial_state.to_scalar("norm_squared").integral.real)
eq = PDE({"ψ": f"I * laplace(ψ)"}) # define the pde
# solve the pde and store intermediate data
storage = MemoryStorage()
eq.solve(initial_state, t_range=2.5, dt=1e-5, tracker=[storage.tracker(0.02)])
# visualize the results as a space-time plot
plot_kymograph(storage, scalar="norm_squared")
def plotky():
plot_kymograph(storage)
def kymograph_pic():
plot_kymograph(storage)
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