def test_steady_state_tracker():
""" test the SteadyStateTracker """
storage = MemoryStorage()
c0 = ScalarField.random_uniform(UnitGrid([5]))
pde = DiffusionPDE()
tracker = trackers.SteadyStateTracker(atol=1e-2, rtol=1e-2, progress=True)
pde.solve(c0, 1e3, dt=0.1, tracker=[tracker, storage.tracker(interval=1e2)])
assert len(storage) < 9 # finished early
def integrate(tfinal, u, u2, udiff, IC):
grid = CartesianGrid([[0, 200]], 200)
field = ScalarField(grid, IC)
storage = MemoryStorage()
eq = libraryPDE(u, u2, udiff) #define PDE with custom parameters
return eq.solve(field,
t_range=tfinal,
dt=0.1,
tracker=storage.tracker(0.1)).data
def test_small_tracker_dt():
"""test the case where the dt of the tracker is smaller than the dt
of the simulation"""
storage = MemoryStorage()
pde = DiffusionPDE()
c0 = ScalarField.random_uniform(UnitGrid([4, 4]), 0.1, 0.2)
pde.solve(
c0, 1e-2, dt=1e-3, method="explicit", tracker=storage.tracker(interval=1e-4)
)
assert len(storage) == 11
def test_pde_time_dependent_bcs(backend):
"""test PDE with time-dependent BCs"""
field = ScalarField(grids.UnitGrid([3]))
eq = PDE({"c": "laplace(c)"}, bc={"value_expression": "Heaviside(t - 1.5)"})
storage = MemoryStorage()
eq.solve(field, t_range=10, dt=1e-2, backend=backend, tracker=storage.tracker(1))
np.testing.assert_allclose(storage[1].data, 0)
np.testing.assert_allclose(storage[-1].data, 1, rtol=1e-3)
def main():
xstep=5
tstep=1
v=73.82/60#Unit:cm/s
tmin=0
tmax=500
bc_T=f'-1*{llambda}/{h}*c.diff(x)+(Piecewise(\
((25+{T_Kelvin}),tt<=20/{v}),\
((25+150/{v}*(tt-20/{v})+{T_Kelvin}),And(20/{v}<tt,tt<=25/{v})),\
((175+{T_Kelvin}),And(25/{v}<tt,tt<=197.5/{v})),\
((175+20/{v}*(tt-197.5/{v})+{T_Kelvin}),And(197.5/{v}<tt,tt<=202.5/{v})),\
((195+{T_Kelvin}),And(202.5/{v}<tt,tt<=233/{v})),\
((195+40/{v}*(tt-233/{v})+{T_Kelvin}),And(233/{v}<tt,tt<=238/{v})),\
((235+{T_Kelvin}),And(238/{v}<tt,tt<=268.5/{v})),\
((235+20/{v}*(tt-268.5/{v})+{T_Kelvin}),And(268.5/{v}<tt,tt<=273.5/{v})),\
((255+{T_Kelvin}),And(273.5/{v}<tt,tt<=344.5/{v})),\
((255-230/{v}*(tt-344.5/{v})+{T_Kelvin}),And(344.5/{v}<tt,tt<=435.5/{v})),\
((25+{T_Kelvin}),true)).subs(tt,t))'
#ture需要小写,f'string'string里面有{}可以传递全局参数参数
grid = CartesianGrid([[0, x_max]], [int(xstep)],periodic=False)
state=ScalarField(grid=grid,data=T_origin+T_Kelvin)
eq = DiffusionPDE(diffusivity=alpha, bc={'value_expression':bc_T},)
storage = MemoryStorage()
eq.solve(state, t_range=(tmin,tmax),dt=1e-3, tracker=storage.tracker(tstep))
def temp_curve():
p=[]
for i in range(int((tmax-tmin)/tstep)):
p.append(storage.data[i][-1])
q=np.asarray(p)
plt.plot(np.linspace(tmin,tmax,int((tmax-tmin)/tstep),endpoint=0),q,label='Temperature')
plt.xlabel('Time')
plt.ylabel('Temprature')
plt.show()
def plotky():
plot_kymograph(storage)
#temp_curve()
plotky()
def test_solvers_simple_ode(scheme, adaptive):
"""test explicit solvers with a simple ODE"""
grid = UnitGrid([1])
field = ScalarField(grid, 1)
eq = PDE({"y": "2*sin(t) - y"})
dt = 1e-3 if scheme == "euler" else 1e-2
storage = MemoryStorage()
solver = ExplicitSolver(eq, scheme=scheme, adaptive=adaptive)
controller = Controller(solver, t_range=20.0, tracker=storage.tracker(1))
controller.run(field, dt=dt)
ts = np.ravel(storage.times)
expect = 2 * np.exp(-ts) - np.cos(ts) + np.sin(ts)
np.testing.assert_allclose(np.ravel(storage.data), expect, atol=0.05)
if adaptive:
assert solver.info["steps"] < 20 / dt
else:
assert solver.info["steps"] == pytest.approx(20 / dt, abs=1)
def main():
#ture需要小写,f'string'string里面有{}可以传递全局参数参数
eq = Heat_trans()
storage = MemoryStorage()
eq.solve(state, t_range=(tmin,tmax),dt=1e-3, tracker=storage.tracker(tstep))
def temp_curve():
p=[]
for i in range(int((tmax-tmin)/tstep)):
p.append(storage.data[i][-1])
q=np.asarray(p)
plt.plot(np.linspace(tmin,tmax,int((tmax-tmin)/tstep),endpoint=0),q,label='Temperature')
plt.xlabel('Time')
plt.ylabel('Temprature')
plt.show()
def plotky():
plot_kymograph(storage)
#temp_curve()
plotky()
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
grid = CartesianGrid([[0, x_max]], [int(xstep)])
state = ScalarField(grid=grid, data=T_origin + T_Kelvin)
eq = PDE(
rhs={'c': f'{alpha}*laplace(c)'},
bc={
'type': 'mixed',
'value': -1 * h / llambda,
'const': bc_T
},
)
#bc_ops={})
storage = MemoryStorage()
eq.solve(state, t_range=(tmin, tmax), dt=1e-3, tracker=storage.tracker(tstep))
def temp_curve():
p = []
for i in range(int((tmax - tmin) / tstep)):
p.append(storage.data[i][-1])
q = np.asarray(p)
plt.plot(np.linspace(tmin, tmax, int((tmax - tmin) / tstep), endpoint=0),
q,
label='Temperature')
plt.xlabel('Time')
plt.ylabel('Temprature')
plt.show()
#!/usr/bin/env python3
from pde import UnitGrid, ScalarField, DiffusionPDE, MemoryStorage, movie_scalar
eq = DiffusionPDE() # define the physics
grid = UnitGrid([16, 16]) # generate grid
state = ScalarField.random_uniform(grid, 0.2,
0.3) # generate initial condition
storage = MemoryStorage() # create storage
tracker = storage.tracker(interval=1) # create associated tracker
eq.solve(state, t_range=2, dt=0.005, tracker=tracker)
# create movie from stored data
movie_scalar(storage, '/tmp/diffusion.mov')
\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")
#!/usr/bin/env python3
from pde import (DiffusionPDE, UnitGrid, ScalarField, MemoryStorage,
PlotTracker, PrintTracker, RealtimeIntervals)
eq = DiffusionPDE() # define the pde
grid = UnitGrid([16, 16]) # generate grid
state = ScalarField.random_uniform(grid, 0.2,
0.3) # generate initial condition
storage = MemoryStorage()
trackers = [
'progress', # show progress bar during simulation
'steady_state', # abort when steady state is reached
storage.tracker(interval=1), # store data every simulation time unit
PlotTracker(show=True), # show images during simulation
# print some output every 5 real seconds:
PrintTracker(interval=RealtimeIntervals(duration=5))
]
eq.solve(state, 10, dt=0.1, tracker=trackers)
storage[0].plot(show=True)
"""
Time-dependent boundary conditions
==================================
This example solves a simple diffusion equation in one dimensions with time-dependent
boundary conditions.
"""
from pde import PDE, CartesianGrid, MemoryStorage, ScalarField, plot_kymograph
grid = CartesianGrid([[0, 10]], [64]) # generate grid
state = ScalarField(grid) # generate initial condition
eq = PDE({"c": "laplace(c)"}, bc={"value_expression": "sin(t)"})
storage = MemoryStorage()
eq.solve(state, t_range=20, dt=1e-4, tracker=storage.tracker(0.1))
# plot the trajectory as a space-time plot
plot_kymograph(storage)
(195+8*tt+273.15,And(233<tt,tt<=238)), \
(235+273.15,And(238<tt,tt<=268.5)), \
(235+4*tt+273.15,And(268.5<tt,tt<=273.5)), \
(255+273.15,And(273.5<tt,tt<=344.5)), \
(255-(230/91)*tt+273.15,And(344.5<tt,tt<=435.5)), \
(25+273.15,And(435.5<tt,tt<=500))).subs(tt,t))'
grid = CartesianGrid([[0, x_max]], [int(xstep)])
state=ScalarField(grid=grid,data=T_origin+T_Kelvin)
eq = PDE(rhs={'c': 'a*laplace(c)'},
bc={'value_expression': bc_T},
consts={'a':alpha})
storage = MemoryStorage()
eq.solve(state, t_range=tmax, tracker=storage.tracker(tstep))
def kymograph_pic():
plot_kymograph(storage)
def temp_curve():
p=[]
for i in range(int(tmax/tstep)):
p.append(storage.data[i][-1])
q=np.asarray(p)
plt.plot(np.linspace(0,tmax,int(tmax/tstep),endpoint=1),q,label='Temperature')
plt.xlabel('Time')
plt.ylabel('Temprature')
plt.show()
"""
Stochastic simulation
=====================
This example illustrates how a stochastic simulation can be done.
"""
from pde import (KPZInterfacePDE, UnitGrid, ScalarField, MemoryStorage,
plot_kymograph)
grid = UnitGrid([64]) # generate grid
state = ScalarField.random_harmonic(grid) # generate initial condition
eq = KPZInterfacePDE(noise=1) # define the SDE
storage = MemoryStorage()
eq.solve(state, t_range=10, dt=0.01, tracker=storage.tracker(0.5))
plot_kymograph(storage)